HAL Id: tel-02171255https://tel.archives-ouvertes.fr/tel-02171255

Submitted on 2 Jul 2019

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Design of exact solutions for the manufacturing of ”vias”using DSA technology

Dehia Ait Ferhat

To cite this version:Dehia Ait Ferhat. Design of exact solutions for the manufacturing of ”vias” using DSA technology.Modeling and Simulation. Université Grenoble Alpes, 2018. English. �NNT : 2018GREAM094�. �tel-02171255�

THÈSE

Pour obtenir le grade de

DOCTEUR DE LA COMMUNAUTÉ UNIVERSITÉ GRENOBLE ALPES

Spécialité : Mathématiques et Informatique

Arrêté ministériel : 25 mai 2016

Présentée par

Dehia AIT FERHAT

Thèse dirigée par Gautier STAUFFER, Professeur, G-INP

préparée au sein du Laboratoire des Sciences pour la Conception, l'Optimisation et la Production de Grenoble dans l'École Doctorale Mathématiques, Sciences et technologies de l'information, Informatique

Conception de solutions exactes pour la

fabrication de "vias" en utilisant la

technologie DSA

Design of exact solutions for the

manufacturing of "vias" using DSA

technology

Thèse soutenue publiquement le 15 octobre 2018, devant le jury composé de :

Monsieur GAUTIER STAUFFER PROFESSEUR, KEDGE BUSINESS SCHOOL - BORDEAUX, Directeur de thèse

Monsieur FRANÇOIS CLAUTIAUX PROFESSEUR, UNIVERSITE DE BORDEAUX, Rapporteur

Monsieur STEPHANE DAUZERE-PERES PROFESSEUR, ECOLE DES MINES DE SAINT-ETIENNE, Rapporteur

Madame NADIA BRAUNER PROFESSEUR, UNIVERSITE GRENOBLE ALPES, Présidente

Monsieur ANDRAS SEBO DIRECTEUR DE RECHERCHE, CNRS DELEGATION ALPES, Examinateur

Madame LILIA ZAOURAR INGENIEUR DE RECHERCHE, CEA PARIS-SACLAY, Examinatrice

Contents

Acknowledgment 4

1 Introduction 61.1 Context of the research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.1 Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.2 Multiple Patterning (MP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.3 Directed Self Assembly (DSA) . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.4 DSA-aware MP problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 Graph theory overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.1 Graph theory definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.3 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Integer programming approach 242.1 ILP formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.1.1 Formulations for the (general) k−path coloring problem . . . . . . . . . . . 252.1.2 Formulations for the (induced) k−path coloring problem . . . . . . . . . . . 292.1.3 Formulations valid for both problems . . . . . . . . . . . . . . . . . . . . . 31

2.2 Benchmark instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.1 Results for industrial graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3.2 Results for grid graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3.3 Discussion of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.4 ILP VS heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Dynamic programming for graphs with bounded treewidth 473.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Treewidth and dynamic programming . . . . . . . . . . . . . . . . . . . . . . . . . 493.3 Dynamic programming for proper 3-colorability . . . . . . . . . . . . . . . . . . . . 51

3.3.1 Dynamic programming using tree decomposition . . . . . . . . . . . . . . . 513.3.2 Dynamic programming using nice tree decomposition . . . . . . . . . . . . 53

3.4 Dynamic programming for 1-path 3-coloring problem . . . . . . . . . . . . . . . . . 543.5 Dynamic programming for induced 2-path 3-coloring problem . . . . . . . . . . . . 553.6 Benchmark instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.7 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2

4 k-matching cover 634.1 Notations and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 From DSA-aware MP to k−matching cover . . . . . . . . . . . . . . . . . . . . . . 644.3 Matching and network flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.1 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.2 Network flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 k-matching cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.4.1 Proof of Theorem 4.5 (first version) . . . . . . . . . . . . . . . . . . . . . . 724.4.2 Proof of Theorem 4.5 (second version) . . . . . . . . . . . . . . . . . . . . 75

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Conclusion and perspectives 77

Contributions 79

Bibliography 83

Appendix A Definitions 84

Appendix B Algorithm for grid graphs 87

Appendix C D-FLAT implementation 89C.1 Input format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89C.2 D-FLAT code for 1-path coloring problem . . . . . . . . . . . . . . . . . . . . . . . 89C.3 D-FLAT code for 2-path coloring problem . . . . . . . . . . . . . . . . . . . . . . . 90

Appendix D CPLEX OPL implementation 93D.1 Data file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93D.2 GFBPNEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93D.3 GFBPNC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96D.4 GFBPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98D.5 ICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100D.6 IFB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102D.7 IDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105D.8 IAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106D.9 Naïve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3

Acknowledgment

The results of the research work outlined in this document have been made possible through nu-merous contributions. To all those who have provided scientific support or friendly influence, I expressmy deepest gratitude.

I would like to thank the members of the jury for agreeing to participate in the review of my the-sis document and defense. Thank you to the reviewers of my manuscript, Stéphane Dauzère-Péres,Professor at the Ecole des Mines de Saint-Etienne, and François Clautiaux, Professor at BordeauxUniversity. I am grateful to them for carefully reading and reviewing my thesis document and for theirinterest in this work.

I would like to thank Nadia Brauner, Professor at the Grenoble Alpes University, for agreeing tobe the president of the jury during my defense.

I would like to express my gratitude to Gautier Stauffer, Professor at KEDGE Business School inBordeaux, for having accepted to supervise this thesis work. Thank you for sharing your knowledge,thank you for all your advice, your availability, your patience and your positive energy that you wereable to bring the entire process. Thank you for your support and encouragement during periods ofdoubt that I experienced.

Many thanks to Vincent Juliard, R&D Manager at Mentor Graphics for offering to work on thisthesis subject, for guiding (following, supporting) me, and for helping me to keep a very good balancebetween the G-SCOP Laboratory and Mentor Graphics company. I also want to thank Andres Torres,Principal Engineer at Mentor Graphics for the time and the energy he always found from the otherside of the world, the USA.

Thank you to András Sebö, Research Director at CNRS Delegation Alpes for our very construc-tive discussions and your contribution on the "k-matching cover" part of this work. And thank you toZoltán Király for bringing your expertise for the same part of the work.

Having done a thesis at a company in collaboration with a laboratory, I was fortunate to have twooffices. Therefore, I would like to thank my colleagues at Mentor Graphics for welcoming me. Thankyou for all the jogging, pilates, hiking or cycling sessions. I could say that I have done a lot of exerciseduring these years of theses, and I am delighted about it.

Thank you to Georgios Manetas and Vivian Bregier for finding the time to reread/review my thesischapters. Thank you for your advice in general. I also would like to thank all my colleagues for theirsupport and good attitude. Karine, Alban, Vivian, Ghislain, Kacem, Joost, Nhung, Flore, Ana-Maria,Eve-Marie, Soraya, Olivier, Michael, Hossam, Alexandre, Ruping, Pavlos, Nikita, Jared, Sid-Ahmed,Joydeep, Germain and many others.

4

Thank you to all the members of the lab who were able to share some of their knowledge and/orencouragement with me, who have always been present, helpful and very nice to me: Matthieu, Lucie,Elodie, Rémi, Lucas, Louis, Matej, Vincent, Zoltán, Nicolas, Gricha, Clément, Kleber, Daniel, Hos-sein, Marie-Jo, Fadila, Kevin and many others.

I want to thank all the people outside the laboratory and the company for supporting me all alongbut also thank you for the various adventures and nice moments we shared together: Ania, Mira, Farid,Amel, Danica, Malaurie, Fio, Nicole, Gwen, Anh Thu, Max, Kahina. A special thanks to Sébastienfor his support, for always listening and giving me advice during the difficult times.

Finally, I would like to thank my parents and family for their support. A special thanks to my twocousins, Nabil and Salim (Sissa), who have always been available, who helped me, guided me andsupported me during these years of thesis and well before that.

5

Chapter 1

Introduction

In this chapter we introduce the context and the problem of our research. First, we present somemain steps in the fabrication of integrated circuits, in particular Very Large Scale Integration (VLSI)circuits. First, we focus on two techniques employed for the fabrication of some extremely densecircuits: Multiple Patterning (MP) and more recently Directed Self Assembly (DSA). We will explainhow DSA can be combined with MP in a process called DSA-aware MP. Second, we show how tomodel MP and DSA-aware MP using graphs. We will then define the problem of our research thatis directly related to the DSA-aware MP process. Finally, we present the state of the art of the DSA-aware MP problem and some variants through a review of the complexity of the problem and thealgorithms appearing in the literature.

1.1 Context of the research

During the fabrication of integrated circuits, a large number of transistors is etched over a siliconwafer (or silicon substrate). A dense network of metal conductors, called « interconnects » is depositedon multiple horizontal layers within dielectric medium material (non-conductive material) on top ofthe transistors. Interconnects provide the electrical current paths among the transistors. Short verticalconductors called « vias » are used to connect horizontal interconnect segments between two differentlayers. Note that via’s definition is important, as we will see in the next sections that our research isdirectly related to it. A depiction of these structures appears in Fig. 1.1 and Fig. 1.2.

Nowadays, an integrated circuit may contain more than one billion transistors and about 15 metal-lic layers, on a surface of 1cm2. The size of certain fabricated features within the transistors are in theorder of 28 nm (i.e. 28.10−9m) and below. There are two main steps for the fabrication of integratedcircuits as shown schematically in Fig. 1.3:

— Circuit design: First, the conception and an abstract description of the circuit is done. Next,the physical implementation of the previous abstract implementation takes place. The physicalimplementation consists in placing the transistors over the available chip area and laying outthe interconnect routing among them, over multiple layers. This provides the geometricaldescription called layout of the circuit system. Several layout layers are defined for onecircuit design.

— Circuit fabrication: Every layer is manufactured following 3 main steps: Coating, lithogra-phy and etching. These steps are described in more details below.

The manufacturing of integrated circuits is done on silicon wafers, each silicon wafer has a diam-eter of 300mm (note that the size of the wafer evolved over time and keeps evolving) and containshundreds of identical integrated circuits placed next to one another. The different steps of the manu-

6

Metal layers

Components (transistors,…)

Vias

Siliconwafer

Figure 1.1 – A zoom on an integrated circuit [39].

Figure 1.2 – A 3D view of « vias » (vertical conductors) connecting horizontal conductors of differentmetal layers.

facturing of integrated circuits are described in detail in [58]. An overview of the main steps is givenin the following:

— Coating: The silicon wafer is prepared to receive the circuit components. Thus, differentlayers of chemical materials are deposited on the silicon wafer, see (a) and (b) in Fig. 1.4.

— Lithography: The features (geometrical shapes) of each layer are drawn on an opaque platecalled a mask (usually made of glass). During lithography, the coated silicon wafer, an opticallens and the mask containing the shapes to be printed on the wafer are stacked on one anotherwith a specific distance between them and a light source is projected onto this system as shown

7

Abstract description Geometrical description

of circuits of circuits

Printing geometrical shapes

on silicon wafer

Circuit design Circuit fabrication

Figure 1.3 – Integrated circuits fabrication flow

in Fig. 1.4 (c). Through this process, features are transferred from the mask to the top layer ofthe silicon wafer where their size is typically reduced by a factor of 4, due to the magnificationof the lens. The transfer of the features on the wafer is done by removing the exposed surfaceof its top layer, due to its photosensitive property, see (d) in Fig. 1.4.

— Etching: After lithography, different chemical steps are used, the transferred features on thetop layer are etched, see (e) in Fig. 1.4.

The silicon wafer runs through multiple cycles of coating, lithography and etching, the manufac-tured circuits are tested directly on the wafer, then sectioned into individual dies, assembled, packagedand tested again. See Fig. 1.5 for a general Intel illustration of the fabrication of integrated circuits aspresented in [39].

1.1.1 Lithography

One of the main issues in circuit manufacturing is miniaturization [20]. The industry wants tomake circuits smaller and smaller, following the trend predicted by Moore [57], in order to extracttheir best qualities which are: small size, speed of calculation and low energy consumption. Miniatur-ization makes circuits tiny and very dense in their number of components.

Lithography is one of the important steps during manufacturing [58]. As explained above, lithog-raphy is used to transfer features i.e. an arrangement of vias (or other shapes to be etched on thewafer) called layout, from a mask to a wafer by exposing the wafer to a light source through the mask.The shapes of the features on the mask are different from their results on the wafer. For instance, viasare represented by squares on the masks and after the mask gets exposed to the light source they areetched on the wafer with a shape resembling cylinders. The wavelength of the light source has animpact on the resolution of the pattern we can engrave: the minimal distance permitted between anytwo features is usually referred to as lithography distance and we denote it by Lithodist . (Note thatthe distance we consider between two features defined by 2 regions f1, f2 in the plane is the euclideandistance i.e. minx∈ f1,y∈ f2 ||x− y||2).

The resolution limit (Lithodist) is dependent on the wavelength of the light source projected on themask which is different from the one that arrives on the top layer of the silicon wafer. This differenceis due to diffraction and if the features are at a distance less than the resolution limit, the transferredfeatures will be different from what is expected. In practice, the resolution limit is reduced by usingmore efficient types of light sources. Also, an improvement on the environment of the etching (e.g.immersion of the silicon wafer in a water before lithography step) as well as an improvement on thechemical layers deposited on the silicon wafer may reduce the resolution limit of a given light source.

Currently, 193 immersion technology is used during the lithography steps with a resolution limit

8

(c) Lithography step

(e) cross section view for etching step

transferred on the resist layer

Features on the mask are

Light source

Mask with features

Lens

(d) cross section view for lithography transfer

(b) cross section view for coating step(a) isometric view of coating step

Layer to be etched

Photosensitive resine layer

Substrate silicon layer

Silicon wafer

Figure 1.4 – Manufacturing steps: Coating, lithography and etching.

9

a. Preparing the silicon wafer for lithography

b. A lithography step is applied several times times

Source of light

Mask

Projection lens

Wafer

c. A zoom on an area on the wafer

d. A zoom on one integrated circuit on the wafer, after several steps of lithography

e. Testing and packaging every integrated circuit

f. Deliver

Multi-Core Processor

Wafer

Loop

Figure 1.5 – An overview of the fabrication steps of integrated circuits [39].10

of 45nm [69], while the next generation of lithography, based on Extreme Ultra Violet (EUV) light,can print features with a resolution of 28nm [40]. However, EUV is not used yet in large productionscale and lithography technologies tend to reach their limit: the industry is looking for other solutions.Multiple patterning is one solution used by the industry to go beyond the resolution limit of a givenlithography process.

1.1.2 Multiple Patterning (MP)

Trying to make integrated circuits more compact and more efficient imposes to increase the densityof the layout features. Therefore, in a given layout we may find two features at distance < Lithodist .In this case we cannot put all the features within one mask and apply a single lithography step. Theidea of MP is to decompose the original layout into sub-layouts such that within each sub-layout thedistance between two features is ≥ Lithodist and then to etch with feasible masks one after the otherto produce the original arrangement (we sometime identify the sub-layout with the mask that is usedto produce it). This decomposition is referred to as layout decomposition.

Different processes of lithography are used for multiple patterning. For instance in double pat-terning, Litho Etch Litho Etch (LELE), Litho Freeze Litho Etch (LFLE) and Self Aligned DoublePatterning (SADP) are the most common processes [56]. These methods extend naturally to triplepatterning or more : we can use a sequence LELELE in triple patterning for example. Conceptually,those methods are similar though and we will not distinguish between these in this study. Neverthe-less, for the purpose of illustration, we will most often refer to a sequence of Litho Etch steps : seeFig. 1.6 for an example of double patterning using the LELE process.

(b) first etching step(a) first lithography step

(c) second lithography step (d) second etching step

mask two

mask one

Figure 1.6 – Double patterning using LELE process.

While multiple patterning may potentially decrease the minimum lithography distance, it substan-tially increases the production time and costs. Indeed, the production of the masks is particularlyexpensive. The cost set of all masks needed for the manufacturing of an integrated circuit reach formillions of dollars. The number of masks corresponding to via layers is important. Hence, it is ofparamount importance to minimize the number of lithography steps needed to produce the via lay-ers, during fabrication. Besides one of the main drawbacks of multiple patterning is the alignment.For instance, suppose that we use double patterning: first, the features on the first patterning step areetched on the silicon wafer; then, the features on the second patterning step have to be aligned with

11

the first set of printed features and then etched on the silicon wafer. When the number of patterningsteps increases, the perfect alignment of features from different masks becomes very hard. This is oneof the reasons why, the number of patterning steps used in the industry is typically reduced to at most4 patterning steps.

The current standard in manufacturing is in fact to use double patterning (DP) in most cases andthen triple patterning (TP) or quadruple patterning (QP) when DP is not feasible. Quadruple pattern-ing allows to manage most practical situations. However reducing from N patterning steps to N− 1patterning steps is regarded as a major challenge in order to maintain a reasonable yield (we will seelater that this is a challenging mathematical question too).

The industry already developed heuristics to solve the multiple patterning problem (which consistin minimizing the number of patterning steps for a given layout) [49], [72], [56]. However, there isan interest for new processes that could be used on top of multiple patterning to reduce the number ofpatterning steps further. In this study we are looking for one such promising process called DirectedSelf Assembly (DSA).

1.1.3 Directed Self Assembly (DSA)

Directed Self Assembly (DSA) [35] is a chemical approach based on di-block copolymer (BCP), acombination of two different structures formed by an attraction between different molecules. Fig. 1.7shows an example of a block copolymer called PS-block-PMMA (Poly-Styrene-block-Poly-MethylMethacrylate). DSA works as follows: a region, called a guiding pattern, is filled with a BCP in arandom state (i.e. an unorganized mixture of different blocks of molecules). After a thermo physicalprocess is triggered (called a microphase separate anneal), the BCP assembles into a periodic ar-rangement of block copolymer structures where the homopolymer blocks align between neighboringmolecules: the periodic structures can be cylinders, lamella or other geometric structures as shown infigure (a) in 1.8. These structures depend on the nature of the block copolymer and the ratio of vol-umes occupied by the two homopolymers. In Fig. 1.8 (a), the lamella structures are obtained when thevolume of PS homopolymer is equal to the volume of PMMA homopolymer, the cylinder structuresare obtained when the volume of PS homopolymer is 30% and the volume of PMMA homopolymeris 70%.

Figure 1.7 – An example of a di-block copolymer in a random state composed by a PS homopolymerand PMMA homopolymer [51].

12

In this work we are interested in periodic cylinder structures: a 2D lattice of cylinders with thefollowing properties: the shortest distance between two cylinders is denoted by L0 (the natural period)and the diameter of the cylinders D0 depend on the di-block copolymer, see figure (b) in Fig. 1.8. It ispossible to stretch the natural period L0 by 10% to 15% and we denote the maximum possible distanceby U0. In our industrial setting, we have U0 ≤ Lithodist , hence, [L0,U0]⊆ [0,Lithodist ].

Combining microphase separate with additional chemical steps to keep the cylinders, we couldtake advantage of the DSA technology to print vias that are at distance ≤ Lithodist . The whole idea incombining DSA and lithography steps is to group into guiding patterns some vias that otherwise couldnot be assigned to the same mask. Lithography is then used to print the guiding patterns and DSA isused to etch the vias that are within the guiding patterns, as shown in figures 1.9 and 1.10.

Figure 1.8 – (a) Different organised structures of the di-block copolymer PS-block-PMMA aftermicro-phase [31]. (b) An organised cylinders structure (di-block copolymer after microphase sepa-rate). The distance center to center of two successive cylinders is the natural period L0 of the di-blockcopolymer [29].

(c) microphase separate, structures forming

(a) lithography and etching steps (b) block copolymer coating

(d) vias etching

mask one

Figure 1.9 – Using DSA process with lithography for vias printing.

13

(b) DSA grouping

(d) Guiding pattern

(c) Lithography mask

(f) Vias etching

(a) Vias layout

of the guiding pattern

shape after etching

(e) Di-block copolymerafter microphase using DSA

(vias at distance ≤ Lithodistare linked by an edge)

Figure 1.10 – An example of DSA process.

Not all sets of vias can be grouped. Indeed DSA imposes design rules on the guiding patterns andthus on the feasible groups. A group of vias is said to be feasible (or valid) if we can print a guidingpattern that will allow the formation of the corresponding vias after the DSA process. Because theguiding pattern will have to be etched using lithography, the lithography technology will also have animpact on the feasible groups.

Feasibility of guiding patterns can be checked through a procedure called DSA flow. Fig. 1.11presents a typical DSA flow that allows to create guiding patterns and verify their feasibility. It goesthrough the following steps:

1. Grouping: select which vias to group into a guiding pattern.

2. Synthesis: define the geometric shape of the guiding pattern that will receive the BCP copoly-mer and the cylinders at the expected vias location.

3. Optical Proximity Correction (OPC): Adds optical corrections to the generated guiding pat-tern in order to get the right shape after lithography step.

4. Verification: verify that the selected vias are printed at the correct position and with the rightshape.

Design rules for guiding patterns were investigated and explicit constraints on feasible via groupswere put forth in [18] and [9]. However, for the moment, there are only few specific shapes of guidingpatterns that were validated following the whole DSA flow. The main certified feasible groups concernstring of vias. We define a string of vias as a sequence (v1, ...,vk) of vias. The following constraintsare imposed for DSA grouping to work appropriately considering both 193 nm immersion and EUVlithography:

14

Design

target

*Compose DSA

Compliant layer

*Compose litho

targetOPC *DSA verification

DSA groupingLitho target Litho mask DSA contour

Figure 1.11 – DSA flow.

1. Distance between two successive vias: For a given BCP with corresponding natural periodL0, the distance center to center between two consecutive vias in a string of via should bewithin the DSA distance range [L0,U0].

2. Number of vias in a group: This number is preferably small. Typically, no more than 4 viasper group are allowed otherwise there is high risk that defects will appear within a group: thecontours of the guiding pattern will not be sufficient to let the BCP self align in the expectedway (vias will not be generated in the right shape and position or the number of vias will differfrom expected). This maximum number might evolve in the future with the advanced researchon BCP capabilities. Nevertheless, studying solutions where the number of vias in a group isrestricted to 2 or 3 is already an interesting problem for the industry, as it can reduce the totalnumber of patterning steps required.

3. Positions of vias in a group: There are conditions related to the positions of vias in a groupwhich are specific to the lithography technology.

(a) 193 nm immersion lithography: vias in a group have to respect manhattan configuration(i.e groups of vias must be aligned and parallel to the X or Y axis). See figure 1.12.

(b) EUV lithography: Every three consecutive vias must form an angle between [135◦,180◦],this angle range may evolve according to EUV research. See figure 1.13 for some exam-ples of valid and invalid EUV strings. Figure 1.13 shows an example of more complicatedshape which is valid under our assumption but that should be verified through the full DSAflow of Fig. 1.11.

1.1.4 DSA-aware MP problem

In this study, we are interested in minimizing the number of lithography steps, for the productionof the vias, as this is what decreases most the production cost. We will distinguish two kinds of masksin such a process : we call a DSA mask a mask that involves single vias and at least a non trivialguiding pattern (at least two vias are grouped in this mask), and a Litho mask a mask that does notinvolve guiding patterns, it contains only single vias at distance≥ Lithodist . Manufacturing constraintsimpose that a DSA mask can use only one type of block copolymer to etch vias within that mask : in-deed all vias in that mask are etched through DSA after all guiding patterns have been printed throughlithography (even single vias of this mask are printed with DSA). It implies in particular that all viashave the same size within such a mask and the valid groups used within that mask must use the sameBCP, and then the same range [L0,U0].

15

180o

180o

180o

Valid groups (Manhathan)

180o180o

180o

Invalid groups (non Manhathan)

y

x

Figure 1.12 – Valid groups for 193nm immersion lithography are horizontal or vertical strings.

There are some studies in the literature that were interested in using different BCPs for differentmasks. However, this would involve additional manufacturing difficulties. In this study we assumethat we use only one BCP type for all DSA masks. A mask is valid if the features (vias or groups ofvias) are at distance ≥ Lithodist . The problem of DSA-aware MP (or hybrid patterning) is to find thebest way to group the vias and to decompose the corresponding set of features into valid masks so asto minimize the total number of masks.

1.2 Graph theory overview

In this section we will model MP and DSA-aware MP as graph coloring problems, we first give afew definitions from graph theory. Other standard definitions can be found in Appendix A. Also, wepresent a state of the art of the different techniques and complexity results appearing in the literature,for the MP problem and the DSA-aware MP problem.

1.2.1 Graph theory definitions

Definition 1.1. (Proper graph coloring). Given a graph G = (V,E), a proper coloring is an assign-ment of colors to all vertices of G where two adjacent vertices receive different colors. The minimumnumber of colors that can be used to color a graph G is called the chromatic number of G and isdenoted by χ(G). Let L be an integer, a graph G is L−colorable if there is a proper coloring of G withL colors.

We consider the two following extensions of proper coloring. For both definitions, we consider agraph G = (V,E), a set F ⊆ E and two integers k and L.

First, a k−path L−coloring is an assignment of L colors to the vertices of G so that the connectedcomponents in each color admit an Hamiltonian path (i.e. a path that visits every vertex of the con-nected component exactly once) with edges in F of length ≤ k. A formal definition is given below.

16

90o

90o 45o 45o 45o

Invalid groups

135o135o

Has to be simulated through DSA flow

180o

180o180o135o

165o

165o

Valid groups

x

y

Figure 1.13 – Presentation of valid and invalid positions of vias in a group for EUV lithography. Morecomplex positions should be validated using DSA flow.

17

For an example, see Fig. 1.14.

Definition 1.2. (k−path L−coloring). Let G = (V,E) be a graph and F ⊆ E. Let k ∈ N. Let L ∈ Nand C = {1, . . . ,L}. Let f : V →C a function. Let Vi = {v ∈V : f (v) = i}. Let Fi = {(u,v) ∈ F : u ∈Vi,v ∈Vi}. Let GF = (V,F) . f is a k−path L−coloring of G if:

— There exists a subset of edges Fj ⊆ Fi such that the subgraph (Vi,Fj) is a disjoint union ofpaths of length at most k in GF .

— There is no edge in E between different connected components of GF [Vi].We call the minimum number of colors needed in a k−path coloring of a graph G the k−path

chromatic number and we denote it by χk(G).

bbbbbb bbbb

bb

bbbb

bb

bbbb

bb

bbbb

bb

bbbb

bb

b b

b

b

b

b

b b

b

b

b

bb

b

b

b

b

b b b

b

b

b

bb

b

b

b

b

b

(a) A graph G = (V,E), (b) 3-path 2-coloring. (c) Nodes in color blue form a path of length 1,

Nodes in color red form a connected

component that has a Hamiltonian path

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

bb

bb

b

[L0, U0]Lithodist

edges in F are in bold.

of length 3.

Figure 1.14 – 3−path 2-coloring example.

Second, an induced k−path L−coloring of G is an assignment of L colors to the vertices of G sothat the graph induced by each color forms a disjoint union of paths of F of length ≤ k. A formaldefinition is given below. For an example, see Fig. 1.16.

Definition 1.3. (Induced k−path L−coloring). Let G = (V,E) be a graph and F ⊆ E. Let k ∈ N. LetL ∈ N and C = {1, . . . ,L}. Let f : V →C a function. Let Vi = {v ∈V : f (v) = i}. Let GF = (V,F). fis an induced k−path coloring of G if:

— GF [Vi] = G[Vi] and it induces a disjoint union of paths of length at most k.— There is no edge of E between different connected components of GF [Vi].

We call the minimum number of colors needed in an induced k−path coloring of a graph G the inducedk−path chromatic number and we note it χk

ind(G).

Note that the 1−path coloring problem is equivalent to the induced 1−path coloring problem.Also, as will be clear in the next section, graphs emerging from our applications are unit disk graphs.

Definition 1.4. (Unit disk graph (UDG)). A UDG is the intersection graph of disks of unit diameterin the plane i.e the vertices represent the unit disks centers and two vertices are adjacent if their disksoverlap. Equivalently, two vertices of a UDG are adjacent if the euclidean distance between the centerof the corresponding disks is less or equal to 2, see Fig. 1.15 for an example.

18

b b

b b

b

b

b

b

b

b

b b

b b

b

b

b

b

b

b

b b

b b

b

b

b

b

b

b

(a)Nodes of the graph

are centers of unit disks

(b)Edges of the graphare pairs of nodes whosedisks overlap

(c)A unit disk graph

Figure 1.15 – A unit disk graph in the plane.

1.2.2 Problem formulation

Given a set of vias in the plane and a minimum lithography distance Lithodist , we define a graphG = (V,E) where V represents the set of vias and E is the set of pairs of vias (vi,v j) such that vi ∈Vand v j ∈ V and the euclidean distance between vi and v j is less or equal to the lithography distance,i.e. dist(vi,v j)≤ Lithodist . Then we present, in the following, how to model MP and DSA-aware MPas graph coloring problems.

Multiple patterning modeling:

In case of multiple patterning, there is a clear one-to-one correspondence between coloring ofthe graph G and mask assignment. Remember, in multiple patterning there are only litho masks.Hence, minimizing the number of litho masks is equivalent to finding the chromatic number of thegraph G which is NP-hard in general. The calculation of distance between vias is done on the squares(vias shapes on the mask). However, the problem is closely related to the problem where vias wouldbe cylinders. In this case, the graph G is a unit disk graph. In [1], it is shown that the L−coloringproblem (i.e. can we color the graph vertices with L colors?) on unit disk graphs remains NP-completeif we fix the number of colors L≥ 3. It means that, in our setting, trying to reduce the number of masksfrom 4 to 3 is a challenging mathematical question.

DSA-aware MP modeling:

We consider the same graph G= (V,E) as defined above and a maximum number of vias per groupk > 0. Given a block copolymer (BCP), with natural period L0 and maximum distance U0 (rememberthat we have [L0,U0] ⊆ [0,Lithodist ]), we define F ⊆ E as the set of pairs of vias (vi,v j) that are atdistance [L0,U0]. Vias that are linked with edges in F are « groupable » and vias that are linked withedges in E\F are « ungroupable ».

When k = 1, DSA-aware MP is equivalent to MP. When k = 2, the problem is equivalent to a1−path coloring problem (or an induced 1−path coloring problem as they are equivalent). Whenk ≥ 3, we consider both the (k− 1)−path coloring and the induced (k− 1)−path coloring models.Indeed, there is still some ambiguity in the industry on the right model when k ≥ 3. We call the firstmodel the general case and the second the induced case. And we study both in the following chapters.Note that we should also in principle add angle restrictions on the top of the pure path coloring models.

19

(b) Define vias at distance≤ Lithodist and within [L0, U0]

(c) A unit disk graph modeling vias layout(a) Set of vias layout in the plane

b b

b b

b

b

b

b

b

b

b b

b b

b

b

b

b

b

b

b b

b b

b

b

b

b

b

b

b b

b b

b

b

b

b

b

b

(d) Proper 3-coloring (e) Induced 2−path 3-coloring (f) Induced 2−path 2-coloring

b

b

b

b

c1

c1

c1

c1

c1

c2

c2

c2

c3

c3

c1

c1

c1

b b

b

b

c2 c2

c2

c3

c3

c1

c1

b

b

bc1

c1

c1

c2

c2

c2

c2

c1

c1

c1

(c) A unit disk graph modeling vias layout(c) A unit disk graph modeling vias layout

Graph G(V,E) and F ⊆ E

Lithodist [L0, U0]

edges in F are in bold

(blue,red,black) (blue,black)

Figure 1.16 – Different ways of selecting the paths gives different number of colors. Nodes within onecolor represent disjoint paths of length at most k where k = 2 (induced 2-path coloring case).

However, in our study we do not add an explicit constraint for the angle restrictions, but this couldeasily be included in our models.

1.2.3 State of the art

In this section, we present the state of the art of the multiple patterning problem, the DSA-awareMP problem and their variants: we review the complexity of the problem, existing algorithms fromgraph theory and electronic design automation (EDA) ones.

Multiple patterning and proper coloring:

As we have already observed, multiple patterning can be modeled as a graph coloring problem. Inwhat follows we give a survey on the complexity of the graph coloring problem and the different algo-rithms that are used in the litterature to solve the problem. Graph coloring is one of the most studiedproblem in graph theory, there are many applications in different domains, for example in timetabling

20

and scheduling [19] [22], frequency assignment [32], and different applications in computer science,see for instance [3].

Karp [45] showed that graph coloring is NP-hard. Then in [28], Garey and Johnson proved thatL−colorability is NP-complete when L = 3 and for any integer L≥ 3. In [23], Pardalos et al presenta bibliographic survey on the proper graph coloring problem where they discuss complexity aspects,problem formulation using Integer Linear Programming, heuristics and algorithms.

A lot of research has been done on polynomial-time approximation algorithms with performanceguarantees [23]. One of the first approximation algorithm for the proper graph coloring was presentedby Johnson [44]. The performance guarantee is O(n/logn) where n is the number of nodes of thegraph. There are some improvements done on this performance guarantee. However, Garey and John-son [28] showed that no polynomial-time algorithm can have a performance guarantee of less thantwo.

Besides, various heuristics and meta-heuristics methods were also used to solve the graph coloringproblem. In [7], Aslan et al presented a survey of heuristics to solve graph coloring problem such asFirst Fit (FF), Recursive Largest First (RLF) or Degrees of Saturation (DSATUR). Simulated Anneal-ing (SA) and the Tabu Search (TS) were also used for the graph coloring problem [23].

The special case of unit disk graph coloring was most studied in communication network ap-plications for the so called « frequency assignment problem » [32]. In [1], Graf et al proved thatL-colorability of unit disk graphs for any fixed integer L≥ 3 is NP-complete. The four-color theoremstates that planar graphs are 4-colorable [4], [5], [6], [62]. Checking if a planar graph is 3-colorableis NP-complete though [28]. In [61], Peeters proved that 3−colorability of planar unit disk graphs isalso NP-complete. Moreover, Peeters introduced a polynomial time approximation algorithm for unitdisk graph coloring with a performance guarantee of 3.

Graph coloring can be solved in polynomial time in some cases. 2−colorability can be checked inpolynomial time by testing if the graph is bipartite (using Breadth First Search). L−colorability can besolved in polynomial time for perfect graphs [30]. For graphs with small treewidth (a treewidth is ameasure of how « tree like » a graph is, the smaller the closer it is) it can also be solved in polynomialtime through dynamic programming [15] (in this case the run time is polynomial in the graph sizeand exponential in the treewidth [15]).

Given a graph G, a natural lower bound on the chromatic number is the size of the maximumclique of G (a clique is a set of nodes such that every node is adjacent to the others, and the size ofthe maximum clique is denoted by ω(G)), i.e. ω(G) ≤ χ(G). In particular, the lower bound on thechromatic number is tight in the case of a perfect graph [12], [54], [55]. In general though, the gapbetween ω(G) and χ(G) can be arbitrarily large, see Zykov [2] and Mycielski [43]. In [17], Brookproved that for any given graph G, the chromatic number of G is at most the maximum degree of G,∆(G), unless G is a complete graph or an odd cycle where the chromatic number is ∆(G)+1.

There are other types of graph coloring, in particular the edge coloring problem is the problem ofcoloring the edges of a graph with a minimum number of colors such that no two edges of a same colorare incident to the same vertex. The minimum number of colors to edge color a graph G is called thechromatic index and it is denoted by χ ′(G). Edge coloring is known to be NP-hard [36]. However, in[46] König proved that for any bipartite graph G, the chromatic index of G is equal to the maximumdegree of G, i.e. χ ′(G) = ∆(G). In [66], Vizing proved that a graph G can be edge-colored with either∆(G) or ∆(G)+1, i.e. ∆(G)≤ χ ′(G)≤ ∆(G)+1. There is a direct equivalence between edge coloring

21

of a graph and vertex coloring of the corresponding line graph (see the definition of a line graph in theAppendix A.9): Let G be a graph and L(G) its line graph then the chromatic index of G is equal tothe chromatic number of L(G), i.e. χ ′(G) = χ(L(G)).

The electronic design automation industry has also developed proprietary heuristics to solve themultiple patterning process, and also exact methods have been presented in the literature for the caseof double patterning and triple patterning, see [49], [72].

DSA-aware MP and k−path L−coloring:

As discussed earlier, DSA-aware MP can be modeled as a general k−path L−coloring problem(or an induced k−path L−coloring problem) on unit disk graphs. In the following, we give a surveyon the complexity of the problems and the different algorithms that are used in the literature to solveDSA-aware MP and a variant of the DSA-aware MP problem called « DSA grouping and mask assign-ment problem ». DSA grouping and mask assignment is a problem that the industry is also interestedin: given a fixed number of masks, assign the vias to groups and masks so as to minimize the numberof conflicts, a conflict appears each time two vias are in different groups of a same color (possibly asingleton) and are below the lithography distances.

In [34], Havet et al introduced an advanced frequency assignment problem which is modeled asa k−improper L−coloring of unit disk graphs where k is an integer, i.e. a coloring of the nodes withL colors, such that no node receives the same color as k+1 or more of its neighbors. Different com-plexity results are presented in [34], in particular it is proved that, in unit disk graphs, k−improperL−coloring is NP-complete for any fixed integers k and L such that k≥ 0 and L≥ 3. They also provedthat, in unit disk graphs, k−improper 2−coloring is NP-complete for any fixed positive integer k. Notethat the k−improper L−coloring of unit disk graphs is equivalent to the k−path L−coloring of unitdisk graph (as defined in 1.2.2) when k = 1 and F = E. Hence, the 1−path L−coloring problemof UDG is NP-complete. Moreover, we became aware recently that the induced k−path L−coloringproblem when F = E, is known in the litterature [41] as the (o,r) path coloring problem, where o isthe order of the path (number of nodes in the path) and r is the number of colors used to color all thenodes of the graph. In [42], Jinjiang proved that the (2,2) path coloring problem is NP-complete andin [41] he showed that the (3,3) path coloring problem is NP-complete. Thus, the induced 1−path2−coloring and the induced 2−path 3−coloring problems are also NP-complete. Although, we donot have a formal reduction, it seems likely that the k−path L−coloring and the induced k−pathL−coloring problems are NP-complete for any L≥ 2 and k ≥ 1.

In [71], Badr et al presented some first heuristics for DSA grouping and mask assignment. Theystudied two sequential approaches: first define the DSA groups and then apply the mask assignmentor the reverse. They showed that the sequential approaches fail to find optimal solutions even forsmall layouts and advocated the solution of considering DSA grouping and mask assignment simulta-neously. In [8], Badr et al presented two other methods used in case of at most 2 vias per DSA group:A simultaneous DSA grouping and mask assignment based on Integer Linear Programming formu-lation and a sequential DSA grouping followed by a mask assignment heuristic based on MaximumCardinality Matching (MCM) which is solvable in polynomial time. In [48], Kuang et al presented anexact method to solve the simultaneous DSA grouping and double patterning, and a heuristic in caseof DSA grouping and triple patterning. Their method makes the assumption that after some graphsimplification most of the subgraphs are planar and they exploit this property. Kuang et al showed thattheir results outperform those presented by Badr et al in [71]. Yang et al in [70] claimed that the pro-posed methods in [48] are difficult to understand, therefore they introduced an alternative approachto solve the sequential DSA grouping and mask assignment, exploiting here again the property that

22

most of the graphs are nearly planar (it is sufficient to delete a few edges from the graph to becomeplanar). The method is based on a meta-heuristic. First, DSA groups are generated using a set packingmethod then a local search method is used for mask assignment. Both methods are encapsulated in amulti-start strategy. Yang et al showed that their results outperform those presented by Badr et al in[71] for the MCM heuristic and they are generally comparable to the results presented by Kuang et alin [48]. In [37], Karageorgos et al presented an algorithm for the DSA grouping and mask assignmentproblem based on exact and heuristic methods: the exact algorithm is applied on very small connectedcomponents (10 to 20 nodes) for a fixed run time; if the best solution is not found then a heuristic isused to find a good solution in a reasonable amount of time.

Mentor Graphics’ EDA tools for DSA-aware double patterning implement a heuristic algorithmknown under the name DFM-DSA-DP. The algorithm is used mainly when at most 2 vias are allowedper DSA group and the number of masks is fixed to two. DFM-DSA-DP is a greedy algorithm thatdefines a « via order » using their position in the plane and then assigns the smallest color availablein this order, while satisfying the grouping and coloring constraints. Mentor is interested in exactmethods to solve the k−path L−coloring problem in order to help assess the quality of the heuristicmethods they develop (and potentially integrate the corresponding algorithms if they are fast enough).

In our study we develop exact approaches for the DSA-aware MP problem. In Chapter 2, wefocus on integer linear programming methods. In Chapter 3, we introduce a dynamic programmingapproach for graphs with bounded tree width. In Chapter 4, we develop a polynomial algorithm in thespecial case of line graphs.

23

Chapter 2

Integer programming approach

Integer linear programming (ILP) is one of the standard exact methods to solve a combinatorialoptimization problem. In an ILP formulation, the objective and the constraints of the problem aredefined as linear functions. For the same problem, several ILP formulations may exist. In this chapter,we present different ILP formulations for the general and induced k−path coloring problem. To thebest of our knowledge there is no existing formulation except one formulation proposed by Havet etal [34] for the 1-improper coloring problem that can be used for the 1-path coloring problem (as theyare equivalent). We thus introduce different formulations and we compare them computationally.

It is known that compact ILP formulations for the graph coloring problem usually contain manysymmetries (i.e. example: for the sequence of vertices 1-2-3-4 and two colors “blue” and “green”these two colorings are symmetric: blue-green-green-green and green-blue-blue-blue). One solutionto this problem is to use a column generation algorithm. To use this algorithm, the exponential sizeILP formulation has to be written implicitly under a very specific format (define a main problem,the so-called « master problem » as well as a sub problem of the original one, the so-called « pricingproblem »). Note that we did not investigate « column generation » type of formulations since in ourcase, the number of colors needed is small ( ≤ 4 or 5) and thus the symmetries are limited: in fact wedid validate this format computationally by doing preliminary tests with BaPCod [38]. BaPCod is a« black-box » implementation of the column generation algorithm.

Most of our formulations are built upon the « naïve » formulation for proper coloring and we thusstart with recalling the latter. Then, we introduce three ILP formulations for the general k−path color-ing problem and four ILP formulations for the induced k−path coloring problem. Finally, we presenta naïve formulation based on the enumeration of all possible feasible paths of length at most k whichis valid for both problems, it only differs in the list of feasible paths.

After presenting all formulations, we compare them on different benchmarks. Two kinds of in-stances are considered, a first group are instances from the industry, a second group are artificialinstances that we designed to resemble the industrial ones. We ran experiments for k = 1 and k = 2 asthis is currently the state of the art in practice (to be able to implement the corresponding DSA-awareMP).

2.1 ILP formulations

Let us first introduce a classic (weak) integer programming formulation for proper coloring. Givenan undirected graph G= (V,E) and a bound L on the number of colors, we consider the following vari-ables for all v ∈V and i ∈ [L] = {1, . . . ,L}:

24

xiv ∈ {0,1} : indicates if node v is colored with color i.

λ i ∈ {0,1} : indicates if color i is chosen.

The following model solves the coloring problem (if L is big enough):

minL

∑i=1

λi (2.1)

∑i∈[L]

xiv = 1 ∀v ∈V (2.2)

xiu + xi

v ≤ λi ∀(uv) ∈ E,∀i ∈ [L] (2.3)

xiv,λ

i ∈ {0,1} ∀i ∈ [L],∀v ∈V (2.4)

The objective function 2.1 minimizes the number of colors needed to color all the vertices. Con-straint 2.2 ensures that each vertex receives exactly one color. Constraint 2.3 forces two adjacentnodes to receive different colors. This is a classic formulation for the graph coloring problem which isknown to be weak in general. Indeed, this formulation presents a lot of symmetries. However, whenthe bound L on the number of colors is small (as it is the case in our application), the formulation isacceptable.

2.1.1 Formulations for the (general) k−path coloring problem

The most natural way of extending the model above to the k−path coloring problem is to keep in-dexing the colors and to introduce variables that allow to represent the structures of the solution withinone color. In proper coloring, the node set within one color forms a stable set (i.e. no two vertices areadjacent). In this new problem, it must form connected components for which a Hamiltonian path (i.ea path that visits every vertex of a connected component exactly once) of length ≤ k (with edges inF) exists. The idea is to model paths using flow variables and conservation constraints (i.e. the sumof the flow outgoing from a vertex is at most the sum of the flow entering to that vertex). The mainchallenge is to ensure that the paths are of length ≤ k. For this constraint, we use the standard trick ofbuilding a « time-expanded network ». This is defined in the first model named a General Flow Basedmodel with Path, Node Edge Coloring (GFBPNEC).

Note that a stable set is now represented as a disjoint union of paths of length ≤ k. Hence, withina color, every node is contained in some path. In all the following formulations we consider: a graphG = (V,E), a set of edges that can be used in a path F ⊆ E, the maximum length of a path k≥ 0, L themaximum number of colors allowed to color nodes in V and NF(v) is the set nodes that are neighborsof vertex v with edges in F .

A general flow based model with path, node and edge coloring (GFBPNEC):

In this model, we assign colors to the selected paths and the nodes and edges associated with thepaths. The variables used in this first formulation are binary variables, they are presented below:

xi(u,v) ∈ {0,1}: indicates if an edge (u,v) ∈ F is chosen in a path with color i ∈ [L].

xi,κ(u,v) ∈ {0,1}: indicates whether an edge (u,v) ∈ F is used as the (κ +1)-th edge in the direction

from u to v, in one of the disjoint paths of color i, where 0 ≤ κ ≤ k− 1 (this explicitly gives anorientation to the path).

yiv,o ∈ {0,1}: indicates if a node v ∈V is in a path starting in o ∈V with color i ∈ [L].

25

yiv,v ∈ {0,1}: indicates if a node v ∈V is the first node of a path starting in v ∈V with color i ∈ [L].

ziv ∈ {0,1}: indicates if a node v ∈V has color i ∈ [L] (hence v is in a path).

λ i ∈ {0,1}: indicates if a color i ∈ [L] is chosen.

The model is presented below:

minL

∑i=1

λi (2.5)

∑u∈NF (v)

xi,κ(v,u) ≤ yi

v,v, if κ = 0,∀i ∈ [L],∀v ∈V (2.6)

∑u∈NF (v)

xi,κ(v,u)− ∑

u∈NF (v)xi,κ−1(u,v) ≤ 0, if κ ≥ 1,∀i ∈ [L],∀v ∈V (2.7)

k−1

∑κ=0

(xi,κ(u,v)+ xi,κ

(v,u)) = xi(u,v), ∀i ∈ [L],∀(u,v) ∈ F (2.8)

yiv,v + ∑

κ=1,...,k∑

u∈NF (v)xi,κ−1(u,v) = zi

v, ∀i ∈ [L],∀v ∈V (2.9)

yiu,o + xi

(u,v)−1 ≤ yiv,o, ∀i ∈ [L],∀(u,v) ∈ F,∀o ∈V (2.10)

yiv,o + xi

(u,v)−1 ≤ yiu,o, ∀i ∈ [L],∀(u,v) ∈ F,∀o ∈V (2.11)

∑o∈V

yiv,o = zi

v, ∀i ∈ [L],v ∈V (2.12)

yiu,o + ∑

o′∈V,o′ 6=oyi

v,o′ ≤ 1, ∀i ∈ [L],∀(u,v) ∈ E,∀o ∈V (2.13)

L

∑i=1

ziv = 1, ∀v ∈V (2.14)

xi,κ(u,v),y

iu,o,x

i(u,v),z

iv ≤ λ

i, ∀i ∈ [L],∀κ,∀(u,v) ∈ F,∀o,u,v ∈V (2.15)

xi,κ(u,v),y

iu,o,x

i(u,v),z

iv,λ

i ∈ {0,1} ∀i ∈ [L],∀κ,∀(u,v) ∈ F,∀o,u,v ∈V (2.16)

All the variables are indexed by a color i ∈ [L]. Let us fix one color i and analyze the structureimposed by the constraints. Constraint 2.6 imposes that if a path starts with an edge from node v ∈Vthen v has to be the first node of the path. Constraint 2.7 imposes that an edge leaving from v canbe the κ-th edge of a path only if there was an edge entering the node v that was the (κ−1)-th edgeof the same path. Constraint 2.8 imposes that an edge (u,v) ∈ F is chosen in color i if and only if itis used in one direction or the other in a path. Constraint 2.9 ensures that a node v ∈ V is chosen incolor i iff either v is the first node of a path in color i or there exists exactly one edge (u,v) ∈ F thatis chosen in a path in color i. Constraint 2.10 and 2.11 identify the path each node belongs to (it canbe seen as a propagation of connectivity). Constraint 2.12 imposes consistency for the value taken byyi

v,o, ∀i ∈ [L],∀v,o ∈V (a vertex is in a path in color i iff it is in color i). Constraint 2.13 ensures thattwo adjacent nodes cannot belong to different paths of the same color. Constraint 2.14 imposes thateach node v ∈ V receives exactly one color. The number of variables in the formulation is (L.k.n2)where n is the number of nodes of the graph.

Observe that the model is well-suited for column generation. However, we tested the approach us-ing BaPCod and the first results discouraged us from investigating further, because as for our instancesCPLEX was performing better (due to the small number of symmetries we assume).

26

A general flow based model with path and node coloring (GFBPNC):

This second model is a variation of GFBPNEC with fewer variables. We assign colors to the se-lected paths and path nodes. All variables are binary variables:

xκ

(u,v) ∈ {0,1} : indicates whether an edge (u,v) ∈ F is used as the (κ +1)-th edge in a path in thedirection from u to v, where 0≤ κ ≤ k−1.

x(u,v) ∈ {0,1} : indicates if the edge (u,v) ∈ F is chosen in a path.yi

v,o ∈ {0,1} : indicates if a node v ∈ V is in a path starting in o ∈ V and is colored with colori ∈ [L].

yv,v ∈ {0,1} : indicates if a node v ∈ V is the first node of a path (then the path is also labeled byv).

λ i ∈ {0,1} : indicates if a color i ∈ [L] is chosen.

The ILP formulation is given below:

minL

∑i=1

λi (2.17)

∑u∈NF (v)

xκ

(v,u) ≤ yv,v, if κ = 0,∀v ∈V (2.18)

∑u∈NF (v)

xκ

(v,u)− ∑u∈NF (v)

xκ−1(u,v) ≤ 0, if κ ≥ 1,∀i ∈ [L],∀v ∈V (2.19)

k−1

∑κ=0

(xκ

(u,v)+ xκ

(v,u)) = x(u,v), ∀(u,v) ∈ F (2.20)

yv,v + ∑κ=1,...,k

∑u∈NF (v)

xκ−1(u,v) = 1, ∀v ∈V (2.21)

yiu,o + x(u,v)−1 ≤ yi

v,o, ∀i ∈ [L],∀(u,v) ∈ F,o ∈V (2.22)

yiv,o + x(u,v)−1 ≤ yi

u,o, ∀i ∈ [L],∀(u,v) ∈ F,o ∈V (2.23)L

∑i=1

yiv,v = yv,v, ∀v ∈V (2.24)

yiu,o + ∑

o′∈V,o′ 6=oyi

v,o′ ≤ 1, ∀i ∈ [L],(u,v) ∈ E,o ∈V (2.25)

L

∑i=1

∑o∈V

yiv,o = 1, ∀v ∈V (2.26)

yiu,o ≤ λ

i, ∀i ∈ [L],∀u,o ∈V (2.27)

xκ

(u,v),x(u,v),yiu,o,λ

i,yv,v ∈ {0,1} ∀i ∈ [L],∀κ,∀(u,v) ∈ F,∀u,v,o ∈V (2.28)

The idea is to move the path description away from the color (notice that variables used to selectpaths are not indexed by color index). We decide and select paths in the graph that covers all nodesthrough constraints 2.18 - 2.21. Then, constraint 2.24 assigns a color to the first node of each path.Constraints 2.22 and 2.23 propagate the color in each path. Constraint 2.26 ensures each node hasa color (and a path in that color). Constraint 2.25 ensures that there is no conflict within each color(there is no edge linking two different paths in the same color), and 2.27 ensures that colors thatare used are accounted for. The number of variables in the GFBPNC formulation is of the order of(k.n2 +L.n2) where n is the number of nodes of the graph.

27

A general flow based model with path coloring (GFBPC):

This third formulation uses even fewer variables than the second one. We assign colors to theselected paths only. We refer to this formulation as General Flow Based model with Path Coloring(GFBPC). All the variables are binary variables:

xκ

(u,v) ∈ {0,1} : indicates whether an edge (u,v) ∈ F is used as the (κ +1)-th edge in a path in thedirection from u to v where 0≤ κ ≤ k−1.

x(u,v) ∈ {0,1} : indicates if an edge (u,v) ∈ F is chosen in a path.yv,o ∈ {0,1} : indicates if a node v ∈V is in a path starting in o ∈V .yi

o ∈ {0,1} : indicates if a path starting in o ∈V is colored with color i ∈ [L].λ i ∈ {0,1} : indicates if a color i ∈ [L] is chosen.

The ILP formulation is given below:

minL

∑i=1

λi (2.29)

∑u∈NF (v)

xκ

(v,u) ≤ yv,v, if κ = 0,∀v ∈V (2.30)

∑u∈NF (v)

xκ

(v,u)− ∑u∈NF (v)

xκ−1(u,v) ≤ 0, if κ ≥ 1,∀v ∈V (2.31)

k−1

∑κ=0

(xκ

(u,v)+ xκ

(v,u)) = x(u,v), ∀(u,v) ∈ F (2.32)

yv,v + ∑κ=1,...,k

∑u∈NF (v)

xκ−1(u,v) = 1, ∀v ∈V (2.33)

yu,o + x(u,v)−1 ≤ yv,o, ∀(u,v) ∈ F,o ∈V (2.34)

yv,o + x(u,v)−1 ≤ yu,o, ∀(u,v) ∈ F,o ∈V (2.35)

∑o∈V

yv,o = 1, ∀v ∈V (2.36)

L

∑i=1

yio = yo,o, ∀o ∈V (2.37)

yio + yi

o′ ≤ 3− (yu,o + yv,o′), ∀(u,v) ∈ E,∀o,o′ ∈V,o 6= o′,∀i ∈ [L] (2.38)

yio ≤ λ

i, ∀i ∈ [L],∀o ∈V (2.39)

xκ

(u,v),x(u,v),yu,o,yio,λ

i ∈ {0,1} ∀i ∈ [L],∀κ,∀(u,v) ∈ F,∀o,u,v ∈V (2.40)

Constraints 2.30- 2.35 are the same constraints as 2.18- 2.23. Constraint 2.36 imposes thatevery node belongs to exactly one path. Constraint 2.37 assigns exactly one color to every selectedpath. Constraint 2.38 ensures that there is no conflict in a color (no edge between two paths of thesame color). The number of constraints and variables in the GFBPC formulation is of the order of(k.n2 +L.n) where n is the number of nodes of the graph.

Note that it is very easy to add restriction regarding forbidden sequences of edges in all the modelsabove. Hence, we could add restrictions on the angle between consecutive edges as it is required inpractice. For instance, if a given degree of an angle formed by the nodes u,v,w (i.e. the value of uvw)is not allowed then we add the following constraints:

28

xκ−1(u,v)+ xκ

(v,w) ≤ 1, ∀κ ≥ 1

xκ−1(w,v)+ xκ

(v,u) ≤ 1, ∀κ ≥ 1

2.1.2 Formulations for the (induced) k−path coloring problem

In this section we consider the induced k−path coloring problem. In this case, the node set withinone color form connected components where each one is a path of length ≤ k and the edges of thepaths are in F . In principle, we could exploit the models for the general case by adding a constraintimposing that the edges that have both extremities of the same color have to be part of color i. How-ever the resulting models would be unnecessarily complicated. In the four models we present now, wetry to exploit further the properties of the induced case.

As defined in the previous chapter, given an undirected graph G = (V,E), a set F ⊆ E and an in-teger k ≥ 0, a partition of V into disjoint subsets V 1, . . . ,V L is an induced k−path L−coloring if G[Vi]induces a disjoint union of paths of GF [Vi] (the subgraph G[Vi] restricted to edges in F) of length ≤ k.In this setting, all we have to decide is whether a vertex is part of Vi or not (we do not have to decidewhether an edges in F is part of color i in contrast with the previous section), and to ensure that thesubgraph G[Vi] induces disjoint path of GF [Vi] of length at most k.

We first propose two formulations for the induced k−path coloring problem for any given value ofk ≥ 0. The first one is somehow similar to the formulation of the traveling salesman problem (TSP).The second one is a flow based formulation similar to GFBPNEC presented in the general problem.Then, we present two formulations that are specific to the 1-path coloring problem (this particularproblem is important for the industry).

An induced formulation with cycle elimination (ICE):

The idea of this formulation is to assign colors to vertices and then to deduce the subgraph G[Vi]by imposing that whenever the 2 extremities of an edge are colored i, then the edge is in G[Vi]. Then,in every color, we check that we have only paths by counting the degree of every node (it should beat most 2) and to verify that no cycle of length ≤ k and no path of length > k appear, we add cycleselimination constraints. All the variables are binary variables :

ziv ∈ {0,1} : Indicates if a node v ∈V is colored with color i ∈ [L].

xi(u,v) ∈ {0,1} : Indicates if the edge (u,v) ∈ F is in the subgraph GF [Vi].

λ i ∈ {0,1} : Indicates whether the color i is chosen.

The formulation is given below:

29

minL

∑i=1

λi (2.41)

L

∑i=1

ziv = 1, ∀v ∈V (2.42)

ziu + zi

v−1 ≤ xi(u,v), ∀i ∈ [L],∀(u,v) ∈ F (2.43)

∑v∈NF (u)

xi(u,v) ≤ 2, ∀i ∈ [L],∀u ∈V (2.44)

∑f∈F [S]

xif ≤ |S|−1, ∀i ∈ [L],∀S⊆V : |S| ≤ k+1 (2.45)

∑f∈F [S]

xif ≤ k ∀i ∈ [L],∀S⊆V : |S|= k+2 (2.46)

ziu + zi

v ≤ 1, ∀i ∈ [L],∀(u,v) ∈ E \F (2.47)

ziu,x

i(u,v) ≤ λ

i ∀i ∈ [L],∀u ∈V,∀(u,v) ∈ F (2.48)

ziu,x

i(u,v),λ

i ∈ {0,1}, ∀i ∈ [L],∀u ∈V,∀(u,v) ∈ F (2.49)

Constraint 2.42 ensures that every node has exactly one color. Constraint 2.43 ensures that if anedge does not receive color i then only one of its two extremities can receive color i. Constraints 2.44,2.45 and 2.46 ensure that the node set that receive color i is an induced disjoint path of G of length atmost k, they impose that every vertex with color i has degree at most 2, there is no cycle of length ≤ kand there is no path of length > k. Constraint 2.47 verifies that no conflict happens by imposing thatno two vertices linked by an edge of E \F receive the same color. The number of constraints of theICE formulation is of the order of (nk) and the number of variables of the order of (L.n2), where n isthe number of nodes of the graph.

An induced flow based model (IFB):

In ICE formulation, the number of constraints as listed in 2.45, 2.46 grows as ∼ nk. We proposeto reduce the size of the model by using the flow based model proposed in the general problem. Theidea is to build paths of length at most k using flow variables and conservation constraints. Then, wedecide only if a vertex has color i. We refer to this formulation as the Induced Flow Based model(IFB). The ILP formulation is given below:

minL

∑i=1

λi (2.50)

∑u∈NF (v)

xi,κ(v,u) ≤ yi

v,v, if κ = 0,∀i ∈ [L],∀v ∈V (2.51)

∑u∈NF (v)

xi,κ(v,u)− ∑

u∈NF (v)xi,κ−1(u,v) ≤ 0, if κ ≥ 1,∀i ∈ [L],∀v ∈V (2.52)

k−1

∑κ=0

(xi,κ(u,v)+ xi,κ

(v,u)) = xi(u,v), ∀i ∈ [L],∀(u,v) ∈ F (2.53)

yiv,v + ∑

κ=1,...,k∑

u∈NF (v)xi,κ−1(u,v) = zi

v, ∀i ∈ [L],∀i ∈ [L],v ∈V (2.54)

30

L

∑i=1

ziv = 1, ∀v ∈V (2.55)

ziu + zi

v−1 ≤ xi(u,v), ∀i ∈ [L],∀(u,v) ∈ F (2.56)

ziu + zi

v ≤ 1, ∀i ∈ [L],∀(u,v) ∈ E \F (2.57)

ziu,x

i(u,v) ≤ λ

i ∀i ∈ [L],∀u ∈V,∀(u,v) ∈ F (2.58)

ziu,x

i(u,v),y

iv,v,x

i,κ(u,v),λ

i ∈ {0,1}, ∀i ∈ [L],∀u,v ∈V,∀(u,v) ∈ F (2.59)

Constraints ( 2.51 - 2.54) define the conservation flow constraints. They are the same as con-straints ( 2.6 - 2.9) in GFBPNEC. Constraint 2.55 imposes that every vertex is exactly in one color.Then we introduce constraints 2.56 and 2.57 that are specific to the induced problem and force dis-joint induced paths in every color. Indeed, constraint 2.56 ensures that if an edge in F is not chosenin a color then its extremities are not in the color (it verifies that we only have induced paths in acolor) and constraints 2.57 ensures that vertices that are linked with an edge in E \F are in differentcolors. The number of constraints and the number of variables in the IFB formulation is of the orderof (L.k.n2) where n is the number of nodes of the graph.

2.1.3 Formulations valid for both problems

We now develop two formulations in the case of 1−path coloring and a « naïve » formulation validfor both.

An induced formulation with degree counting (IDC):

In this formulation, we simplify the ICE formulation by exploiting that each vertex in a path isnow of degree 1. In that case, we impose constraint 2.63 instead of 2.44, 2.45, 2.46.

minL

∑i=1

λi (2.60)

L

∑i=1

ziv = 1, ∀i ∈ [L],∀v ∈V (2.61)

ziu + zi

v−1 ≤ xi(u,v), ∀i ∈ [L],∀(u,v) ∈ F (2.62)

∑v∈NF (u)

xi(u,v) ≤ 1, ∀i ∈ [L],∀u ∈V (2.63)

ziu + zi

v ≤ 1, ∀i ∈ [L],∀(u,v) ∈ E \F (2.64)

ziu,x

i(u,v) ≤ λ

i ∀i ∈ [L],∀u ∈V,∀(u,v) ∈ F (2.65)

ziu,x

i(u,v),λ

i ∈ {0,1}, ∀i ∈ [L],∀u ∈V,∀(u,v) ∈ F (2.66)

The number of constraints and the number of variables in the IDC formulation is of the order of(L.n2) where n is the number of nodes of the graph.

An induced formulation with adjacency counting (IAC):

In this formulation we reduce the number of variables and constraints of the IDC formulation bysubstituting the constraints zi

u + ziv−1≤ xi

(u,v) and ∑v∈NF (u) xi(u,v) ≤ 1 with ∑v∈NF (u) zi

v ≤ ziu +(1− zi

u) ·|NF(u)|. This imposes that each vertex in a color has at most 1 neighbor of the same color without

31

using edge variables. This is equivalent to having a path of length ≤ 1. The formulation is givenbelow:

minL

∑i=1

λi (2.67)

L

∑i=1

ziv = 1, ∀i ∈ [L],∀v ∈V (2.68)

∑v∈NF (u)

ziv ≤ zi

u +(1− ziu) · |NF(u)|, ∀i ∈ [L],∀u ∈V. (2.69)

ziu + zi

v ≤ 1, ∀i ∈ [L],∀(u,v) ∈ E \F (2.70)

ziu ≤ λ

i ∀i ∈ [L],∀u ∈V (2.71)

ziu,λ

i ∈ {0,1}, ∀i ∈ [L],∀u ∈V (2.72)

(2.73)

The number of constraints and the number of variables of the IAC formulation is of the order of(L.n), where n is the number of nodes of the graph.

Observe that the IDC and IAC formulations are valid for both the general and the induced 1−pathcoloring problem as they are equivalent.

A « naïve » formulation:

Finally, we present a very intuitive formulation for small values of k which is valid for both vari-ants of the k−path coloring problem. Let P be the set of all feasible paths of length at most k havingedges in F (all the constraints needed to be set on the paths can be easily considered: induced paths,path lengths, angle restriction, etc). In case of general k−path coloring we refer to this formulation as« general path listing (GPL) » and in case of induced k−path coloring we refer to it as « induced pathlisting (IPL) ». In this formulation we use the following binary variables:

λ i ∈ {0,1} : Indicates if a color i is chosen.xi

p ∈ {0,1} : Indicates if a path p ∈P is colored with color i ∈ [L].The formulation is given below:

minL

∑i=1

λi (2.74)

L

∑i=1

∑p∈P:v∈p

xip = 1, ∀v ∈V (2.75)

xip1+ xi

p2≤ 1, ∀i ∈ [L],∀p1, p2 : p1 6= p2 and ∃u ∈ p1,v ∈ p2 s.t. (u,v) ∈ E

(2.76)

xip ≤ λ

i, ∀i ∈ [L],∀p ∈P (2.77)

xip,λ

i ∈ {0,1}, ∀i ∈ [L],∀p ∈P (2.78)

(2.79)

Constraint 2.75 imposes that each node v ∈ V is in exactly one path and one color. Constraint2.76 imposes that for two paths p1 and p2 such that there exists an edge (u,v) ∈ E where node u is inp1 and node v is in p2 then the two paths have to receive different colors. (Again, we do not care too

32

much about symmetries due to the small L value in practice). The number of variables of the naïvemodel is of the order (L.nk) and the number of its constraints in of the order (L.n2k) where n is thenumber of nodes of the graph.

2.2 Benchmark instances

In our study we use two instance groups. The first group comes from industrial layouts of viasavailable from Mentor Graphics. They are displayed in Table 2.1: « |V | » is the number of nodes,« |E| » is the number of edges, « |F | » is the number of edges that can be assigned to a path, « Den-sity » is the density of the graph calculated as follows: 2|E|

|V |(|V |−1) , « ω(G) » is the largest clique of thegraph, « ∆(G) » is the maximum degree of the graph, « χ(G) » is the chromatic number of the graph.The second group of instances, which we call grid graphs, was designed to resemble the industrialones as follows: We consider the points in a plane with coordinates (x,y) ∈ N2, where x ∈ {1, . . . ,n},y ∈ {1, . . . ,n} and n a positive integer. We create what we call a « grid graph » by setting a valuefor n, and setting the graph nodes to be points we choose with a fixed probability p = 0.5 (we couldchoose different probabilities to vary the density of the graphs). An edge is created between everytwo vertices having an euclidean distance at most two as shown in Fig. 2.1. Our grid graphs are morecomplex than the industrial instances. However, it is useful to test all the formulations with them inorder to see their limits.

u v4

v1

v2

v3

uu

(a) Node u is at distance 1 (b) Node u is at distance 2

from each v1, v2, v3, v4 from each node v1, v2, v3, v4(c) Node u is at distance

√2

from each node v

b

b

b

b b b b b

b

b

b b

bb

b

v3

v2

v1

v4

v1

v2

v4

v3

Figure 2.1 – All adjacent vertices are at distance ≤ 2.

It is hard to extract precise characteristics of industrial layout, because, the rules used to designevery layout are different depending on the lithography technology and the end use of the layout.However, the vias are usually positioned on a « regular grid ». A natural question that may arise aboutthe characteristics of our « grid graphs » is the following: Is the graph coloring an easy problem on thegrid graphs? After some research and discussions with Dr Louis Esperet, we did not find an intuitiveanswer. However, we could show that the grid graphs are not perfect (we know that there is a polyno-mial time algorithm that finds the chromatic number of any perfect graph and, if G is a perfect graphthen ω(G) = χ(G)). To show that grid graphs are not perfect it is sufficient to find an example whereω(G) 6= χ(G). We present two grid graphs showing that the size of the maximum clique is differentfrom the chromatic number. Let’s consider the grid graphs shown in Fig. 2.2 and Fig. 2.3. We can seethat ω(G) = 2 and χ(G) = 3, and ω(G) = 3 and χ(G) = 4, respectively. We leave for future researchthe question of the complexity for this type of instances.

In Table 2.2, we present the characteristics of 36 connected grid graphs that we have generated.We have selected the edges in F as follows: Let u,v ∈ V , if the euclidean distance between u and v

33

b b

b

b

bb

b

b b

b

b

bb

b

b

b

b

b

b

b

b

ω(G) = 2 and χ(G) = 3.

distance = 1

C7

Figure 2.2 – An example showing that in general grid graphs are not perfect: a grid graph mightcontain a C7 and then the maximum clique size is 2 and the chromatic number is 3 (ω(G) 6= χ(G)).

b

b b

b

b

b

b

b

b

b

b

b

bb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b b

b

b

b

b

b

b

b

b

b

bb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

distance = 1

ω(G) = 3 and χ(G) = 4.

b

b

Figure 2.3 – Another example showing that a grid graph G is not perfect in general, in this examplethe maximum clique size is 3 and the chromatic number is 4 (ω(G) 6= χ(G)).

is ≤√

2 then the edge (u,v) ∈ F , this corresponds to L0 =12 Lithodist and U0 =

√2L0 (remember, in

this case Lithodist = 2). Detailed steps on the grid graphs generation are presented in Algorithm 1 inAppendix B.

2.3 Numerical results

The naïve GPL and IPL formulations, were implemented using Cplex Python API (Cplex 12.6.3and Python 3.4). The Networkx Python library was in particular useful in order to list all possiblepaths. All the other formulations GFBPNEC, GFBPNC, GFBPC, IDC, AC, IFB

All the formulations were implemented using the CPLEX OPL 12.6.3. The corresponding codesare given in Appendix D. In the case of naïve GPL and IPL formulations, the list of all possiblepaths is generated using PYTHON 3.4, then given as a set of data in OPL. The tests were done on amachine with an Intel(R) Xeon(R) CPU E5-2640 2.60 GHz and a memory of 529GB, and enforcinga time limit of one hour. In the following, we present the results obtained for the two classes of graph,industrial and grid graphs. Currently, the technology allows essentially to group at most 2 or 3 vias.Hence, we focus on the general and induced k−path coloring problem when k = 1 and k = 2. In tables2.4, 2.5, 2.7, 2.8, 2.10, 2.11, 2.13, 2.14 we present 4 columns: column « Obj » is the value ofthe objective function returned by Cplex, i.e. the best solution returned by the solver: it is an optimal

34

Graph |V | |E| |F | Density ω(G) ∆(G) χ(G)G_1 54 56 52 0.0391 3 4 3G_2 61 86 85 0.0469 3 5 3G_3 96 111 105 0.0243 3 4 3G_4 119 142 136 0.0202 3 4 3G_5 137 167 160 0.0179 3 5 3G_6 159 196 188 0.0156 3 5 3G_7 173 224 216 0.0150 3 5 3G_8 382 396 339 0.0054 3 4 3G_9 993 1009 927 0.0020 3 4 3G_10 1937 1996 1812 0.0010 3 4 3

Table 2.1 – Graphs from via layouts (Mentor Graphics instances).

Graph |V | |E| |F | Density ω(G) ∆(G) χ(G)G_100 42 105 72 0.12 5 9 5G_101 46 98 65 0.09 4 7 4G_102 46 109 71 0.1 5 8 5G_103 114 306 210 0.04 5 10 5G_104 102 250 171 0.04 5 8 5G_105 111 302 202 0.04 5 9 5G_106 188 517 351 0.02 5 9 5G_107 199 533 362 0.02 5 10 5G_108 210 599 412 0.02 5 9 5G_109 331 958 650 0.01 5 11 5G_110 303 820 561 0.01 5 11 5G_111 286 725 482 0.01 5 9 5G_112 462 1425 952 0.01 5 11 5G_113 449 1299 876 0.01 5 11 5G_114 436 1214 812 0.01 5 11 5G_115 623 1800 1225 0.009 5 10 5G_116 624 1821 1221 0.009 5 11 5G_117 604 1691 1112 0.009 5 11 5G_118 811 2316 1547 0.007 5 11 5G_119 764 2090 1400 0.007 5 11 5G_120 840 2560 1733 0.007 5 11 5G_121 1022 3000 2012 0.005 5 11 5G_122 944 2541 1705 0.005 5 11 5G_123 991 2836 1894 0.005 5 11 5G_124 1248 3614 2396 0.004 5 11 5G_125 1266 3678 2456 0.004 5 11 5G_126 1247 3650 2444 0.004 5 12 5G_127 1513 4382 2927 0.003 5 11 5G_128 1503 4355 2892 0.003 5 12 5G_129 1557 4753 3191 0.003 5 12 5G_130 1785 5178 3439 0.003 5 11 5G_131 1763 5055 3391 0.003 5 11 5G_132 1805 5316 3547 0.003 5 12 5G_133 2166 6225 4179 0.002 5 12 5G_134 2091 6086 4083 0.002 5 11 5G_135 2102 6114 4105 0.002 5 12 5

Table 2.2 – Connected random grid graphs where nodes are chosen with probability P = 0.5.

35

value if the solution is found in less than one hour, « Time » is the CPU time in seconds spent byCplex to find the solution, « Gap Cpx » is the percentage gap returned by Cplex i.e. the differencebetween the best lower bound and the « Obj » value, « Gap Rt » is the percentage root gap betweenthe objective solution « Obj » and the objective of the relaxed ILP formulation noted R (i.e. R is theobjective value of the formulation when the variables belong to the interval [0,1]), it is calculated asfollows: Gap Rt = Ob j−R

R × 100. Table entries with the symbol « \ » denote tests where the runtimefor the solver to setup the formulation exceeded the one hour limit.

2.3.1 Results for industrial graphs

Results for k = 1:

In this experiment, we fix k = 1. We are thus interested in the 1−path coloring problem. Theresults for the general 1−path coloring problem are presented in Table 2.4 and the results for theinduced 1−path coloring problem are presented in Table 2.5. The plots in Table 2.3 show the evolutionof the runtime in terms of the graph size (number of nodes) for general and induced problems. Wecan see that the formulations GFBPNEC, GFBPNC and GFBPC are of limited performance: they canfind the minimum number of colors for graphs with at most 382 nodes but no solution is found bythe solver for larger graphs after one hour. Formulation GFBPC is the worst formulation comparedto GFBPNEC, GFBPNC even if it has less constraints and variables. It can solve instances with 119nodes but it did not found a solution to an instance with 61 nodes. Formulation GPL had the bestoverall performances, being able to solve all the industrial instances in less than 6 sec. Coming rightbehind, the formulations of the induced problem IDC, IAC, IFB and IPL can find the optimal solutionfor all the industrial instances within a few seconds, (the largest instance has 1937 nodes). Observethat formulation ICE could not solve the two last instances not because of the runtime spent to solvethe problem (we can see that it can solve the other instances in few seconds), but due to the time spentto setup the formulation. Note that the general 1-path coloring and induced 1-path coloring are exactlythe same problems. Hence, the results of the formulations GPL and IPL are the same in that case.

Results for k = 2:

In this experiment, we fix k = 2. For the general 2−path coloring problem, the results are presentedin Table 2.7 and for the induced 2−path coloring problem the results are presented in Table 2.8. Theplots in Table 2.6 present runtime in terms of the graph size (number of nodes) for both problems.Note that the formulations IDC, IAC do not appear because they are specific to k = 1. Similar tok = 1, formulation GFBPC has worst performance than GFBPNEC, GFBPNC, although, their resultsare close and limited. Formulation GPL has the best runtime overall formulations for the general caseand can solve all instances to the optimal within a few seconds. For the induced problem, formulationsIFB and IPL solved all instances to the optimal within a few seconds. We can see that the runtime ofIPL is slightly faster than that of IFB. Formulation ICE solved five instances in a few seconds, but itstruggles to solve larger instances. It is again the time spent to setup the formulation that increasesdramatically with graph size. When k = 2, the paths listed for the general problem are slightly differentfrom those of the induced problem. However, the runtime spent by the formulations GPL and IPL arevery close.

2.3.2 Results for grid graphs

Results for k = 1:

In this experiment, we study the generated grid graphs and we fix k = 1. In the case of general1−path coloring, the results are presented in Table 2.10 and in the case of induced 1−path coloring in

36

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,0000

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

Number of nodes

Tim

e(s

ec)

GFBPNECGFBPNCGFBPC

GPL

ICE

IDCIACIFBIPL

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,0000

5

10

15

20

25

30

Number of nodes

Tim

e(s

ec)

A zoom on the plots having a runtime of at most 30sec.

GPLICEIDCIACIFBIPL

Table 2.3 – Plots of the runtime with respect to the size of the graphs for the general and inducedformulations where k = 1 on industrial instances.

Graph GFBPNEC GFBPNC GFBPC GPLName |V | Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap

Cpx Rt Cpx Rt Cpx Rt Cpx RtG_1 54 2 11.75 0 0 2 23.74 0 33.33 2 37.3 0 40 2 0.03 0 100G_2 61 2 19.95 0 0 2 115.65 0 66.66 3 3601.66 33.33 62.96 2 0.18 0 0G_3 96 2 48.95 0 0 2 116.78 0 50 2 1321.8 0 0 2 0.21 0 0G_4 119 2 50.12 0 0 2 159.8 0 0 3 3611.55 33.33 33.33 2 0.16 0 0G_5 137 2 73.1 0 0 2 142.45 0 40 3 3620.64 33.33 62.5 2 0.17 0 0G_6 159 2 116.94 0 0 2 1533.29 0 66.66 3 3622.43 33.33 60 2 0.31 0 0G_7 173 2 154.74 0 0 2 2662.67 0 75 3 3620.67 58.33 58.33 2 0.4 0 0G_8 382 2 1336.88 0 0 3 3608.68 33 33.33 \ \ \ \ 2 0.56 0 0G_9 993 \ \ \ \ \ \ \ \ \ \ \ \ 2 0.83 0 100G_10 1937 \ \ \ \ \ \ \ \ \ \ \ \ 3 5.15 0 0

Table 2.4 – General 1−path coloring results for industrial instances.

37

Graph ICE IDC IAC IFB IPLName |V | Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap

Cpx Rt Cpx Rt Cpx Rt Cpx Rt Cpx RtG_1 54 2 0.04 0 0 2 0.28 0 0 2 0.26 0 0 2 0.17 0 0 2 0.03 0 100G_2 61 2 0.58 0 0 2 0.2 0 0 2 0.2 0 0 2 0.25 0 0 2 0.18 0 0G_3 96 2 0.26 0 0 2 0.1 0 0 2 0.12 0 0 2 0.12 0 0 2 0.21 0 0G_4 119 2 0.44 0 0 2 0.41 0 0 2 0.04 0 0 2 0.17 0 0 2 0.16 0 0G_5 137 2 0.58 0 0 2 0.68 0 0 2 0.39 0 0 2 0.22 0 0 2 0.17 0 0G_6 159 2 0.94 0 0 2 0.48 0 0 2 0.04 0 0 2 0.44 0 0 2 0.31 0 0G_7 173 2 1.58 0 0 2 0.6 0 0 2 0.18 0 0 2 0.58 0 0 2 0.4 0 0G_8 382 2 12.31 0 0 2 0.67 0 0 2 0.47 0 0 2 1.28 0 0 2 0.56 0 0G_9 993 \ \ \ \ 2 1.38 0 0 2 1.53 0 0 2 6.38 0 0 2 0.83 0 100G_10 1937 \ \ \ \ 3 26.06 0 33.33 3 4.01 0 33.33 3 22.38 0 33.33 3 5.15 0 16.66

Table 2.5 – Induced 1−path coloring results for industrial instances.

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,0000

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

Number of nodes

Tim

e(s

ec)

GFBPNECGFBPNCGFBPC

GPLICEIFBIPL

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,0000

5

10

15

20

25

30

35

Number of nodes

Tim

e(s

ec)

GPLICEIFBIPL

Table 2.6 – Plots of the runtime with respect to the size of the graphs for the general and inducedformulations where k = 2 on industrial instances.

38

Graph GFBPNEC GFBPNC GFBPC GPLName |V | Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap

Cpx Rt Cpx Rt Cpx Rt Cpx RtG_1 54 2 22.36 0 37.5 2 53.94 0 47.9 2 487.59 0 64.70 2 0.15 0 100G_2 61 2 883 0 50 2 1166.07 0 75 2 3601.62 50 66.39 2 0.48 0 100G_3 96 2 157.41 0 50 2 107.29 0 50 2 591.06 0 0 2 0.49 0 100G_4 119 2 142.23 0 0 2 159.27 0 0 3 3608.28 33.33 33.33 2 0.54 0 100G_5 137 2 171.9 0 50 2 3600.77 37.50 50 3 3611.79 66.67 66.66 2 0.84 0 100G_6 159 2 2045.19 0 50 2 3601.03 50 75 3 3612.27 75 75 2 0.79 0 100G_7 173 2 3601.52 33.33 50 3 3601.07 66.67 89 3 3622.06 75 75 2 1.48 0 100G_8 382 3 3607.9 33.33 33.33 3 3600.9 33.33 33.33 \ \ \ \ 2 0.69 0 100G_9 993 \ \ \ \ \ \ \ \ \ \ \ \ 2 1.87 0 100G_10 1937 \ \ \ \ \ \ \ \ \ \ \ \ 2 3.69 0 100

Table 2.7 – General 2−path coloring results for industrial instances.

Graph ICE IFB IPLName |V | Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap

Cpx Rt Cpx Rt Cpx RtG_1 54 2 0.37 0 0 2 0.23 0 0 2 0.06 0 100G_2 61 2 0.63 0 0 2 0.16 0 0 2 0.52 0 100G_3 96 2 4.02 0 0 2 0.26 0 0 2 0.31 0 100G_4 119 2 9.39 0 0 2 0.29 0 0 2 0.59 0 100G_5 137 2 15.29 0 0 2 0.35 0 0 2 0.59 0 100G_6 159 \ \ \ \ 2 0.39 0 0 2 0.83 0 100G_7 173 \ \ \ \ 2 0.51 0 0 2 1.23 0 100G_8 382 \ \ \ \ 2 1.45 0 0 2 0.83 0 100G_9 993 \ \ \ \ 2 8.55 0 0 2 1.62 0 100G_10 1937 \ \ \ \ 2 30.21 0 0 2 5.39 0 100

Table 2.8 – Induced 2−path coloring results for industrial instances.

Table 2.11. The corresponding plots are given in Table 2.9.

In the general 1−path coloring, the formulation GPL is the best formulation. It can solve all the36 instances in less than 41 sec. Formulation GFBPNEC solves instances of at most 111 nodes. For-mulation GFBPNC solves instances of at most 46 nodes and formulation GFBPC solves instancesof 42 nodes. Both formulations are struggling to find any solution for instances with more than 46nodes. It is interesting to note that decreasing the number of variables from formulation GFBPNECto GFBPNC and GFBPC did not help to solve more instances.

In the induced 1−path coloring, all the induced formulations except ICE solved all the instancesto the optimal. IPL formulation has the best performance, solving all instances to the optimal in lessthan 45 sec. Next follows the IAC formulation with a worst case runtime of 80 sec. Formulations IDCand IFB are 3 time slower than IAC. Formulation IFB is comparable to IDC till the number of nodes1503 then beyond 1503 nodes the runtime of IFB oscillates for two specific instances, the 1513 nodesinstance and 1785 nodes instance. The worst formulation is ICE, it can solve the 12 first instances over36 in few seconds but then the time spent to setup up the formulation for the other instances growsagain dramatically.

Results for k = 2:

For the general 2−path coloring, the results are presented in Table 2.13 and in the case of induced2−path coloring in Table 2.14. The plots in Table 2.12 are the runtime with respect to graph size forgrid graphs.

For the general 2−path coloring, formulation GFBPNEC find optimal solutions for very smallinstances with at most 46 nodes then finds a feasible solution in one hour for instances of at most 286nodes. For graphs exceeding 286 nodes no solution is found. Formulations GFBPNC and GFBPChave the worst performances without giving any optimal solution. In terms of a feasible solution,GFBPNC finds one for instances of at most 286 nodes and GFBPC for instances of at most 210 nodes.

39

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,2000

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

Number of nodes

Tim

e(s

ec)

GFBPNECGFBPNCGFBPC

GPLICEIDCIACIFBIPL

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,2000

50

100

150

200

250

300

350

400

450

Number of nodes

Tim

e(s

ec)

A zoom on the plots with at most 450sec.

GPLICEIDCIACIFBIPL

Table 2.9 – Runtime vs the size of the graphs for the general and induced formulations where k = 1on grid graphs.

40

Graph GFBPNEC GFBPNC GFBPC GPLName |V | Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap

Cpx Rt Cpx Rt Cpx Rt Cpx RtG_100 42 3 83.73 0 33.33 3 542.2 0 66.66 3 1657.18 0 44.08 3 0.18 0 0G_101 46 3 28.44 0 33.33 3 371.98 0 66.66 3 3600.22 33.33 50 3 0.1 0 11.11G_102 46 3 28.84 0 33.33 3 157.55 0 77.77 3 3600.37 33.33 33.33 3 0.12 0 0G_103 114 4 1115.9 0 50 4 3600.36 50 75 4 3632.7 50 50 4 0.31 0 16.66G_104 102 3 210.55 0 33.33 4 3622.63 50 50 4 3608.21 50 50 3 0.44 0 0G_105 111 4 1209.88 0 50 4 3601.37 50 75 4 3610.45 57.95 57.95 4 0.31 0 16.66G_106 188 4 3600.04 49.04 50 4 3601.41 73.86 75 4 3647.33 100 100 4 0.54 0 20.83G_107 199 4 3600.03 29.33 50 4 3601.03 68.75 75 \ \ \ \ 4 0.5 0 19.73G_108 210 4 3600.04 31 50 4 3600.93 75 75 \ \ \ \ 4 0.75 0 22.91G_109 331 4 3600.12 50 50 4 3601.75 75 75 \ \ \ \ 4 1.29 0 16.66G_110 303 4 3600.07 43.53 50 4 3601.67 83.33 83.33 \ \ \ \ 4 1.15 0 15G_111 286 4 3600.05 36.23 50 4 3601.11 50 50 \ \ \ \ 4 0.52 0 16.66G_112 462 \ \ \ \ \ \ \ \ \ \ \ \ 4 1.81 0 100G_113 449 \ \ \ \ \ \ \ \ \ \ \ \ 4 1.7 0 12.5G_114 436 \ \ \ \ \ \ \ \ \ \ \ \ 4 1.81 0 15G_115 623 \ \ \ \ \ \ \ \ \ \ \ \ 4 2.5 0 16.66G_116 624 \ \ \ \ \ \ \ \ \ \ \ \ 4 2.44 0 16.07G_117 604 \ \ \ \ \ \ \ \ \ \ \ \ 4 1.66 0 14.58G_118 811 \ \ \ \ \ \ \ \ \ \ \ \ 4 4.69 0 12.5G_119 764 \ \ \ \ \ \ \ \ \ \ \ \ 4 4.41 0 16.66G_120 840 \ \ \ \ \ \ \ \ \ \ \ \ 4 12.18 0 10G_121 1022 \ \ \ \ \ \ \ \ \ \ \ \ 4 4.94 0 15.62G_122 944 \ \ \ \ \ \ \ \ \ \ \ \ 4 4.94 0 12.5G_123 991 \ \ \ \ \ \ \ \ \ \ \ \ 4 3.39 0 12.5G_124 1248 \ \ \ \ \ \ \ \ \ \ \ \ 4 6.65 0 12.5G_125 1266 \ \ \ \ \ \ \ \ \ \ \ \ 4 6.05 0 12.5G_126 1247 \ \ \ \ \ \ \ \ \ \ \ \ 4 5.69 0 12.14G_127 1513 \ \ \ \ \ \ \ \ \ \ \ \ 4 7.5 0 15G_128 1503 \ \ \ \ \ \ \ \ \ \ \ \ 4 20 0 10G_129 1557 \ \ \ \ \ \ \ \ \ \ \ \ 4 35.72 0 11.36G_130 1785 \ \ \ \ \ \ \ \ \ \ \ \ 4 39.52 0 12.5G_131 1763 \ \ \ \ \ \ \ \ \ \ \ \ 4 8.92 0 15G_132 1805 \ \ \ \ \ \ \ \ \ \ \ \ 4 39.51 0 11.66G_133 2166 \ \ \ \ \ \ \ \ \ \ \ \ 4 44.42 0 12.5G_134 2091 \ \ \ \ \ \ \ \ \ \ \ \ 4 40.88 0 12.5G_135 2102 \ \ \ \ \ \ \ \ \ \ \ \ 4 40.17 0 10

Table 2.10 – General 1−path coloring results for grid graphs instances.

Graph ICE IDC IAC IFB IPLName Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap

Cpx Rt Cpx Rt Cpx Rt Cpx Rt Cpx RtG_100 3 0.23 0 33.33 3 0.23 0 33.33 3 0.11 0 33.33 3 0.24 0 33.33 3 0.18 0 0G_101 3 0.16 0 33.33 3 0.22 0 33.33 3 0.08 0 33.33 3 0.2 0 33.33 3 0.1 0 11.11G_102 3 0.33 0 33.33 3 0.17 0 33.33 3 0.13 0 33.33 3 0.2 0 33.33 3 0.12 0 0G_103 4 0.99 0 50 4 1.14 0 50 4 0.69 0 50 4 2.25 0 50 4 0.31 0 16.66G_104 3 0.59 0 33.33 3 0.78 0 33.33 3 0.19 0 33.33 3 0.51 0 33.33 3 0.44 0 0G_105 4 1.63 0 50 4 1.89 0 50 4 0.67 0 50 4 1.9 0 50 4 0.31 0 16.66G_106 4 14.83 0 50 4 4.74 0 50 4 5.77 0 50 4 74.25 0 50 4 0.54 0 20.83G_107 4 5.01 0 50 4 5.92 0 50 4 1.38 0 50 4 8.08 0 50 4 0.5 0 19.73G_108 4 9.04 0 50 4 6.94 0 50 4 3.11 0 50 4 103.26 0 50 4 0.75 0 22.91G_109 4 16.33 0 50 4 14.53 0 50 4 3.08 0 50 4 6.07 0 50 4 1.29 0 16.66G_110 4 11.93 0 50 4 8.83 0 50 4 2.5 0 50 4 9.25 0 50 4 1.15 0 15G_111 4 19.42 0 50 4 7.73 0 50 4 1.64 0 50 4 36.82 0 50 4 0.52 0 16.66G_112 \ \ \ \ 4 22.42 0 50 4 5.02 0 50 4 10.91 0 50 4 1.81 0 100G_113 \ \ \ \ 4 23.78 0 50 4 3.45 0 50 4 178.95 0 50 4 1.7 0 12.5G_114 \ \ \ \ 4 16.42 0 50 4 4.78 0 50 4 6.96 0 50 4 1.81 0 15G_115 \ \ \ \ 4 44.58 0 50 4 8.08 0 50 4 30.57 0 50 4 2.5 0 16.66G_116 \ \ \ \ 4 45.56 0 50 4 6.51 0 50 4 31.88 0 50 4 2.44 0 16.07G_117 \ \ \ \ 4 27.94 0 50 4 6.36 0 50 4 31.37 0 50 4 1.66 0 14.58G_118 \ \ \ \ 4 52.22 0 50 4 10.32 0 50 4 38.28 0 50 4 4.69 0 12.5G_119 \ \ \ \ 4 37.91 0 50 4 13.86 0 50 4 38.3 0 50 4 4.41 0 16.66G_120 \ \ \ \ 4 45.75 0 50 4 16.35 0 50 4 48.11 0 50 4 12.18 0 10G_121 \ \ \ \ 4 97.2 0 50 4 16.01 0 50 4 59.83 0 50 4 5.94 0 15.62G_122 \ \ \ \ 4 59.43 0 50 4 12.47 0 50 4 24.44 0 50 4 4.94 0 12.5G_123 \ \ \ \ 4 111.71 0 50 4 20.17 0 50 4 79.23 0 50 4 3.39 0 12.5G_124 \ \ \ \ 4 136.09 0 50 4 22.24 0 50 4 62.35 0 50 4 6.65 0 12.5G_125 \ \ \ \ 4 130.59 0 50 4 79.01 0 50 4 138.1 0 50 4 6.05 0 12.15G_126 \ \ \ \ 4 113.55 0 50 4 26.93 0 50 4 108.04 0 50 4 5.69 0 12.14G_127 \ \ \ \ 4 175.81 0 50 4 47.64 0 50 4 1688.13 25 50 4 7.5 0 15G_128 \ \ \ \ 4 153.12 0 50 4 41.46 0 50 4 141.35 0 50 4 20 0 10G_129 \ \ \ \ 4 287.74 0 50 4 42.23 0 50 4 160.94 0 50 4 35.72 0 11.36G_130 \ \ \ \ 4 233.83 0 50 4 51.54 0 50 4 627.12 0 50 4 8.92 0 12.5G_131 \ \ \ \ 4 222.55 0 50 4 51.54 0 50 4 195.13 0 50 4 39.52 0 15G_132 \ \ \ \ 4 258.81 0 50 4 55.94 0 50 4 396.43 25 50 4 39.51 0 11.66G_133 \ \ \ \ 4 365.59 0 50 4 62.04 0 50 4 183.66 0 50 4 44.42 0 12.5G_134 \ \ \ \ 4 374.15 0 50 4 63.53 0 50 4 220.32 0 50 4 40.88 0 12.5G_135 \ \ \ \ 4 283.35 0 50 4 74.54 0 50 4 184.93 0 50 4 40.17 0 10

Table 2.11 – Induced 1−path coloring results for grid graphs instances.

41

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,2000

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

Number of nodes

Tim

e(s

ec)

GFBPNECGFBPNCGFBPC

GPLICEIFBIPL

Table 2.12 – Runtime vs the size of the graphs for the general and induced formulations where k = 2on grid graphs.

Formulation GPL has the best overall performance, solving 26 out of 36 to the optimal and for therest, a feasible solution is returned within one hour.

For the induced 1−path coloring, formulation ICE solves the small instances with at most 114nodes to the optimal in few seconds, for larger instances no solution is found, not even a feasibleone. Formulations IFB and IPL returned either an optimal solution or a feasible solution for all theinstances. When the value of k grows, the instability of the runtime for both formulations is morepronounced. However, the formulation IPL finds more optimal solutions than IFB.

2.3.3 Discussion of the results

In Table 2.15, we present the percentage of instances solved to the optimal for every ILP formula-tion and every class of graphs (industrial instances and grid graphs) when k = 1 and k = 2.

The percentage of the solved instances, for the general k−path coloring in both cases k = 1 andk = 2, shows that formulation GFBPC is the worst formulation comparing to GFBPNEC and GF-BPNC, it solves very few instances. Reducing the number of variables did not help the solver to findbetter solutions. GPL formulation is the best formulation, it solves all the instances from the industrywhen k = 1 and k = 2, it solves all the grid graphs when k = 1, then solves less grid graphs when kstarts to grow.

In the case of induced 1−path coloring, all the induced formulations except ICE solved all theinstances to the optimal. In case of induced 2−path coloring, both IFB and IPL formulations are com-parable, they solved the industrial instances to the optimal. They also solved all the grid graphs to theoptimal when k = 1, however, they start to loose performance when the value of k starts to increase.The IPL formulation solved slightly more grid graphs instances than IFB formulation.

As a conclusion, in both general and induced k−path coloring, somewhat surprisingly the most

42

Graph GFBPNEC GFBPNC GFBPC GPLName |V | Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap

Cpx Rt Cpx Rt Cpx Rt Cpx RtG_100 42 3 2906.96 0 66.65 3 3600.16 33.33 66.65 3 3467.29 33.33 99.09 3 1.67 0 33.33G_101 46 3 423.81 0 66.62 3 3600.07 44.44 66.66 3 3600.9 63.33 99.25 3 0.46 0 33.33G_102 46 3 927.3 0 66.65 3 3600.15 44.54 89.58 3 3606.55 33.33 99.14 3 0.72 0 33.33G_103 114 4 3600.91 50 71.24 4 3600.35 50 75 4 3610.88 72.77 97.73 3 5.53 0 13.33G_104 102 4 3600.47 50 49.99 3 3601.93 33.33 100 4 3614.13 50 97.73 3 2.51 0 23.81G_105 111 3 3600.22 55.35 66.66 4 3601.06 75 74.98 4 3608.96 73.15 96.54 3 3.29 0 15.87G_106 188 4 3600.57 75 74.92 4 3602.55 75 74.96 4 27455.54 100 93.68 3 17.37 0 16.66G_107 199 4 3604.32 75 74.95 4 3601.61 75 100 4 3814.89 100 95.38 3 24.44 0 14.28G_108 210 4 3601.26 75 74.89 4 3601.76 75 74.93 4 3600 75 100 3 49.18 0 16.66G_109 331 4 3621.53 75 75 4 3601.49 75 75 \ \ \ \ 3 104.86 0 12.5G_110 303 4 3601.39 75 75 4 3601.98 87.5 87.5 \ \ \ \ 3 57.49 0 7.93G_111 286 4 3622.24 50 50 4 3601.6 50 50 \ \ \ \ 3 21.43 0 16.66G_112 462 \ \ \ \ \ \ \ \ \ \ \ \ 3 160.68 0 7.40G_113 449 \ \ \ \ \ \ \ \ \ \ \ \ 3 192.39 0 9.37G_114 436 \ \ \ \ \ \ \ \ \ \ \ \ 3 95.4 0 14.14G_115 623 \ \ \ \ \ \ \ \ \ \ \ \ 3 241.19 0 11.11G_116 624 \ \ \ \ \ \ \ \ \ \ \ \ 3 300.81 0 11.11G_117 604 \ \ \ \ \ \ \ \ \ \ \ \ 3 788.08 0 11.02G_118 811 \ \ \ \ \ \ \ \ \ \ \ \ 3 1618.05 0 8.64G_119 764 \ \ \ \ \ \ \ \ \ \ \ \ 3 406.18 0 14.16G_120 840 \ \ \ \ \ \ \ \ \ \ \ \ 3 849.29 0 7.69G_121 1022 \ \ \ \ \ \ \ \ \ \ \ \ 3 951.39 0 12.79G_122 944 \ \ \ \ \ \ \ \ \ \ \ \ 3 267.95 0 11G_123 991 \ \ \ \ \ \ \ \ \ \ \ \ 3 2413.69 0 8.33G_124 1248 \ \ \ \ \ \ \ \ \ \ \ \ 3 2542.76 0 9.15G_125 1266 \ \ \ \ \ \ \ \ \ \ \ \ 3 2390.67 0 9.04G_126 1247 \ \ \ \ \ \ \ \ \ \ \ \ 4 3600.21 25 31.16G_127 1513 \ \ \ \ \ \ \ \ \ \ \ \ 4 3600.07 30.88 32.5G_128 1503 \ \ \ \ \ \ \ \ \ \ \ \ 4 3600.08 25 29.24G_129 1557 \ \ \ \ \ \ \ \ \ \ \ \ 4 3600.12 25 30.26G_130 1785 \ \ \ \ \ \ \ \ \ \ \ \ 4 3600.07 25 32.69G_131 1763 \ \ \ \ \ \ \ \ \ \ \ \ 4 3600.07 25 33.82G_132 1805 \ \ \ \ \ \ \ \ \ \ \ \ 4 3600.1 25 30.90G_133 2166 \ \ \ \ \ \ \ \ \ \ \ \ 4 3600.11 25 31G_134 2091 \ \ \ \ \ \ \ \ \ \ \ \ 4 3600.1 31.53 32G_135 2102 \ \ \ \ \ \ \ \ \ \ \ \ 4 3600.13 25 30.75

Table 2.13 – General 2−path coloring results for grid graphs instances.

Graph ICE IFB IPLName |V | Obj Time Gap Gap Obj Time Gap Gap Obj Time Gap Gap

Cpx Rt Cpx Rt Cpx RtG_100 42 3 0.45 33 33.33 3 0.46 0 33.33 3 0.7 0 11.11G_101 46 3 0.41 0 33.33 3 0.48 0 33.33 3 0.49 0 22.22G_102 46 3 0.47 0 33.33 3 0.36 0 33.33 3 0.52 0 16.66G_103 114 3 9.83 0 33.33 3 2.27 0 33.33 3 2.64 0 8.64G_104 102 3 5.17 0 33.33 3 1.36 0 33.33 3 1.44 0 16.66G_105 111 3 8.39 0 33.33 3 1.44 0 33.33 3 1.64 0 0G_106 188 \ \ \ \ 3 3.91 0 33.33 3 14.26 0 11.11G_107 199 \ \ \ \ 3 4.54 0 33.33 3 8.67 0 8.69G_108 210 \ \ \ \ 3 7.19 0 33.33 3 18.08 0 11.11G_109 331 \ \ \ \ 3 26.29 0 33.33 3 55.52 0 7.40G_110 303 \ \ \ \ 4 459.92 0 50 4 193.85 0 25.78G_111 286 \ \ \ \ 3 6.55 0 33.33 3 5.11 0 0G_112 462 \ \ \ \ 3 44.81 0 33.33 3 26.97 0 0G_113 449 \ \ \ \ 4 478.09 0 50 4 92.13 0 25G_114 436 \ \ \ \ 3 26.91 0 33.33 3 102.52 0 3.03G_115 623 \ \ \ \ 3 75.85 0 33.33 3 29.52 0 0G_116 624 \ \ \ \ 3 110.36 0 33.33 3 40.94 0 0G_117 604 \ \ \ \ 3 98.83 0 33.33 3 29.88 0 0G_118 811 \ \ \ \ 3 133.72 0 33.33 3 164.17 0 0G_119 764 \ \ \ \ 3 156.19 0 33.33 3 47.88 0 0G_120 840 \ \ \ \ 4 3600.08 25 50 4 100.98 0 24.31G_121 1022 \ \ \ \ 3 347.3 0 33.33 3 200.98 0 0G_122 944 \ \ \ \ 3 124.51 0 33.33 3 34.3 0 0G_123 991 \ \ \ \ 4 1489.58 0 33.33 3 333.71 0 0G_124 1248 \ \ \ \ 4 3600.04 37.50 50 3 691.67 0 0G_125 1266 \ \ \ \ 4 3600.12 25 50 4 487.79 0 25G_126 1247 \ \ \ \ 4 3600.04 37 50 4 3600.24 25 25G_127 1513 \ \ \ \ 4 3600.08 50 50 3 1259.28 0 25G_128 1503 \ \ \ \ 4 3600.09 44 50 4 28.28 0 23.27G_129 1557 \ \ \ \ 4 3600.1 45 50 4 910.58 0 25G_130 1785 \ \ \ \ 4 212.56 0 50 4 3600.04 25 25G_131 1763 \ \ \ \ 4 3600.08 50 50 4 3600.17 0 25G_132 1805 \ \ \ \ 4 3600.06 46 50 4 3600.1 25 25G_133 2166 \ \ \ \ 4 175.99 0 50 4 3600.04 25 25G_134 2091 \ \ \ \ 4 3600.1 43.75 50 4 3600.07 25 25G_135 2102 \ \ \ \ 4 3600.22 25 50 4 1693.82 0 24.28

Table 2.14 – Induced 2−path coloring results for grid graphs instances.

43

Path length k = 1 k = 2Formulation grid graphs industrial graphs grid graphs industrial graphsGFBPNEC 16.66% 80% 8.33% 60%GFBPNC 8.33% 70% 0 % 40%GFBPC 2.77% 20% 2.77% 20%GPL 100% 100% 72.22% 100%ICE 33.33% 80% 16.66% 50%IDC 100% 100% \ \IAC 100% 100% \ \IFB 100% 100% 69.44 % 100%IPL 100% 100% 83.33% 100%

Table 2.15 – Percentage of solved instances to the optimal in less than one hour by each formulationamong 36 grid graph instances and 10 industrial instances.

naïve formulations GPL and IPL that list all possible paths of length at most k are the best formulations.However, they solve less instances when it comes to grid graphs with k = 2. Hence, we may expectthat when the value of k grows and the grid graphs become more dense, these formulations willpresent limitations, an analysis would be interesting but we did not investigate this further as for theindustrial application it would have little impact with the current technology limit. For the generalk−path coloring problem, reducing the number of variables in the formulation GFBPC comparing toGFBPNEC and GFBPNC, gave a worst performance.

2.3.4 ILP VS heuristic

The heuristic called DFM-DSA-DP developed by Mentor Graphics works for double patterningonly, in other words, the number of masks is fixed to two then vias are assigned to one mask or anotherfollowing the DSA-aware MP constraints as much as possible. After assigning all the vias to the twomasks, it may happen that some vias will be in conflict. A conflict, or violation, occurs every time theinter-via distance, in the same mask, is not in the DSA range [L0,U0]. The heuristic tries to minimizethe « number of violations ». From a graph theory point of view, the heuristic tries to color as manynodes as possible of a graph using two colors. The number of violations is then the number of noncolored nodes. In practice, the main case study is when at most two vias are allowed per DSA group.Therefore, we will present a comparison between the DFM-DSA-DP heuristic with at most two viasper DSA group and an ILP formulation in case of 1−path 2−coloring. We choose to compare theDFM-DSA-DP to the IAC (induced adjacency counting) formulation. For this study, we adapt theIAC formulation as follows:

— We run the formulation with a fixed set of 2 colors.— As we want to color as many nodes as possible with 2 colors:• Instead of forcing every node to receive exactly one color, we force every node to receive

at most one color.• We change the objective function: we minimize the number of non colored nodes.

Table 2.16 presents a comparison between DFM-DSA-DP and the modified IAC formulation forthree graphs from industrial vias layouts. In addition, we present the results of the proper coloring toshow how DSA-aware MP may help to reduce the number of violations. All the graphs are alreadyknown to be 3 colorable. Every graph has several connected components. Therefore, the ILP formu-lations are applied on every connected component in sequence (only one thread is used). Table 2.16presents four main columns, the first column « Graph » is divided into two sub-columns, « Name »and « # nodes » that give the name and the number of nodes for every graph, respectively. The sec-ond, third and fourth columns are the models « Proper coloring », « DFM-DSA-DP » and « IAC »,respectively. In every model are two columns, « # Violation » is the minimum number of violations

44

Graph Proper coloring DFM-DSA-DP IACName # nodes # Violation Time(sec) # Violation Time(sec) # Violation Time(sec)graph_1 110086 515 1883.3 0 < 1 0 1451.85graph_2 104950 573 1840.9 0 < 1 0 1152.28graph_3 997 96 8.64 0 < 1 0 9.49

Table 2.16 – A comparison between DFM-DSA-DP heuristic vs IAC ILP formulation on 3 graphsfrom the industry where number of colors = 2 and k = 1.

DFM DSA DP: ILP formulation:

Puts more than 2 vias in a group Uses 2 colors following all the constraints

path of length > 1

Figure 2.4 – DFM-DSA-DP vs ILP IAC formulation for 1−path 2−coloring case

and « Time(sec) » is the runtime in seconds spent by the model.

The results show that the heuristic DFM-DSA-DP solved all the violations using DSA as well asthe IAC formulation did. However, while testing the two models DFM-DSA-DP and IAC on otherinstances, we found a very simple case, presented in Fig. 2.4, where DFM-DSA-DP returned twoviolations and IAC formulation returned zero violation. Our ILP formulation helped to assess theresults returned by the heuristic and figure out one of the sensitive cases.

2.3.5 Conclusion

Clearly, the computational results of our formulations showed that the general k−path coloringproblem is harder to solve than the induced k−path coloring problem (except the formulation GPLthat works very well for very small values of k). This might be explained by the fact that, as the valueof k increases the general k−path coloring problem can be seen as a mix of two hard problems: thetraveling salesman problem in order to find the k−paths and a coloring problem in order to assigncolors to the vertices (or assign colors to the paths only then derive the colors of the nodes). Whenthe value of k is very small (k = 1 or k = 2), we were expecting closer results between the general andthe induced formulations. Our expectations were only proved true on the naïve formulations (GPLand IPL), which showed comparable results for both problems (general and induced). All the otherformulations showed a very large gap between the formulations ability to solve the general problemcomparing to the induced one.

There are certainly several opportunities left to improve the formulations, however, we havedemonstrated solutions for the problem of practical interest to the industry, that of the induced 1-path

45

coloring problem. Our proposed formulations solved all industrial instances within a few seconds.Moreover, the formulations returned reasonable results for the induced 2−path coloring problem.

46

Chapter 3

Dynamic programming for graphs withbounded treewidth

In the literature, there are algorithms that can solve some NP-hard problems in polynomial timewhen restricted to graphs with bounded treewidth. For instance, the proper graph coloring problemor the weighted independent set problem [15]. These algorithms are based on dynamic programming.After some analysis on our industrial instances, we found that the corresponding graphs are indeedgraphs with small treewidth. In this chapter, we want to exploit this property and we are thus interestedin solving the k−path coloring problem on graphs with bounded treewidth. First, we give some defini-tions (tree decomposition, treewidth). Then, we present the standard dynamic programming approachand the main steps of this standard approach for the proper 3-coloring problem. Finally, we adaptit to our k−path coloring problem and we present some experimental results for the case of inducedk−path coloring when k = 1 and k = 2 on industrial instances.

3.1 Definitions

The tree decomposition was introduced for the first time by Robertson and Seymour [60]. Aformal definition is given below.

Definition 3.1. (Tree decomposition [60]). Let G = (V,E) be a graph. A tree decomposition of G isa pair (X ,T ) where T is a tree and X = {X1, . . . ,Xn} is the set of nodes of T , called bags such that∀i ∈ {1, . . . ,n}, Xi ⊆V and the three following conditions are verified:

1. ∪i∈{1,...,n}Xi =V .

2. ∀(u,v) ∈ E,∃Xi ∈ X such that u ∈ Xi and v ∈ Xi.

3. ∀u ∈V , the bags containing u is a connected sub-tree of T .

A tree decomposition is characterized by a width. The width of a tree decomposition is its largestbag size minus one. The treewidth of a graph G is the minimum width over all possible tree decom-position of G. An example of a graph and a tree decomposition is presented in Fig. 3.1 with anillustration of properties of the tree decomposition.

Definition 3.2. (Rooted tree decomposition). A rooted tree decomposition is a tree decompositionwhere a bag is chosen as a root and the edges of the tree are oriented in the direction of the root bagi.e. there is a directed path from every bag of the tree decomposition to its root bag. If (X ,T ) is a treedecomposition, we denote by (X ,

−→T ) the corresponding oriented tree decomposition.

For each bag Xi ∈ X , the bags that are linked to Xi with edges toward Xi are called children bagsof Xi and Xi is called the parent bag. The union of the bags in the subtree of T rooted at Xi are noted

47

b

b b

b b

b

b b

a

bc

d

e f g h

a,b,c

c,f

b,c,f

b,e,f

g,h

d,f,g

c,d,f

(a) Graph G(V,E)(b) A tree decomposition (X,T ) of G: Condition (1)

b

b b

b b

b

b b

a

bc

d

e f g h

a,b,c

c,f

b,c,f

b,e,f

g,h

d,f,g

c,d,f

(e) Condition (3) is satisfied: (f) The set of all bags containing node f form a sub-tree

b

b

b b

b b

b

b b

a

bc

d

e f g h

a,b,c

c,f

b,c,f

b,e,f

g,h

d,f,g

c,d,f

(c) Condition (2) is satisfied: (d) The extremities of every edge are contained in at least

b

b

is satisfied as ∪Xi = V

one common bag

Consider for example the node f ∈ V

Consider for example the edge (a, b) ∈ E

Figure 3.1 – An example of a tree decomposition and its properties.

48

Xi

Vi

Xr

Xj

Figure 3.2 – A rooted tree decomposition structure. Xr is the root bag of the tree. Xi is a bag in thetree. Vi is the union of nodes in all bags that belong to the sub-tree rooted at Xi in the rooted treedecomposition.

Leaf Introduce Forget Join

v, u, w

u, wv, u, w

u, w

v

v, u, w

v, u, w

v, u, w

Figure 3.3 – The 4 types of bags in a nice tree decomposition: Leaf, introduce, forget and join bag.

Vi, and Gi is the subgraph of G restricted to nodes in Vi. See figure Fig 3.2 for an illustration of arooted tree decomposition.

In [47], Kloks proved that a tree decomposition can be converted to a nice tree decomposition inlinear time, conserving the same treewidth. We will see later that using a nice tree decomposition ismore suitable for the implementation of dynamic programming algorithms.

Definition 3.3. (Nice tree decomposition). Let G = (V,E) be a graph. A nice tree decomposition is arooted tree decomposition of G where every bag of the tree has at most two children and every bag isone of the four bag types:

— Leaf: A bag Xi is a leaf bag if it has no children and contains only one vertex v ∈ V , i.e.|Xi|= 1.

— Introduce: A bag Xi is an introduce bag if it has exactly one child bag noted X j such that avertex is added to X j i.e. Xi = X j ∪{v} where v ∈V and v 6∈Vj.

— Forget: A bag Xi is a forget bag if it has exactly one child bag noted X j such that Xi = X j\{v}.— Join: A bag Xi is a join bag if it has exactly two children noted X j1 and X j2 such that Xi =

X j1 = X j2.

3.2 Treewidth and dynamic programming

In [63], Arnborg et al have proved that deciding if the treewidth of a graph G is at most w wherew ≥ 0 is NP-complete. However, there are good heuristics to determine a tree decomposition of a

49

given graph G with a small width. Different heuristics for the computation of a tree decompositionare presented in [14] and [59]. Moreover, Arnborg et al [63] showed that for every fixed value of w,there is a linear-time algorithm that finds a tree decomposition of width w (if it exists).

There are algorithms that solve NP-hard problems in polynomial time when restricted to graphswith a treewidth at most w. The complexity of the algorithms are usually polynomial in the size of thegraphs and exponential in the size of the treewidth. This implies that having a small width (even ifnot optimal), while determining a tree decomposition of a given graph G, is of paramount importance.The algorithms that solve problems restricted to bounded treewidth have, generally, the followingstructure:

— First, they compute a tree decomposition with a small width (not necessarily optimal treewidth)then they transform it to a rooted nice tree decomposition.

— Second, they apply dynamic programming on the rooted nice tree decomposition. The mainsteps of the algorithm are: For every bag of the nice tree decomposition compute a table. Everytable gives a list of partial feasible solutions to the problem restricted to the subgraph formedby the set of vertices of its corresponding bag. The table of every bag is computed dependingon the type of the bag (leaf, introduce, forget, join) and it extends a combination of partialfeasible solutions of its children. This is done in a bottom-up way.

— Finally, the solution of the problem to the whole graph can be found in the table of the root bag.

In [21], Courcelle has showed that a large class of problems can be solved in linear time whenrestricted to graphs with bounded treewidth. Courcelle’s theorem states that every graph problem thatcan be formulated in monadic second order logic (MSOL) can be solved in linear time when restrictedto graphs with bounded treewidth. Monadic second order logic as interpreted in graphs is a formu-lation of a property of a graph using logical connectors (∧, ∨), quantification on vertices and setsof vertices (∀v ∈ V , ∃v ∈ V , ∀V ′ ⊆ V ), quantification on edges and sets of edges (∀e ∈ E, ∃e ∈ E,∀E ′ ⊆ E) and the adjacency relation between vertices. An example of a MSOL formula for the proper3-coloring is given in the following for a graph G = (V,E):

∃C1,C2,C3 ⊆V , (∀v ∈V (v ∈C1∨ v ∈C2∨ v ∈C3)) ∧(∀u,v ∈V ) edge(u,v)→ (¬(u ∈C1∧ v ∈C1)∧¬(u ∈C2∧ v ∈C2)∧¬(u ∈C3∧ v ∈C3)).

The formula can be red as: There exist three sets of colors C1,C2,C3 ⊆ V . Every node v ∈ V isin at least one subset C1 or C2 or C3, and for every edge (u,v) ∈ E, nodes u and v are not in the samesubset C1 or C2 or C3. (In other words, the nodes u and v must be in different sets of colors C j wherej ∈ {1,2,3}).

Before starting to apply dynamic programming on k−path L−coloring for graphs with boundedtreewidth, we can verify first that we can express it with a MSOL formula which already proves thatthe problem is then polynomial in graph with bounded treewidth. We show this for the induced casebut the logic would be similar for the general case. We consider L sets of colors C1, . . . ,CL, and avertex v ∈V has to belong to at least one of the sets C j where j ∈ {1, . . . ,L}. In addition, we expressthat every node in every color C j is part of a path of length 0 or 1 or . . . or k. Every path length isreferred to with a predicate noted path_i(C j,v) where the vertex v is in color C j and in a path of lengthi. The corresponding MSOL is given in the following:

∃C1,C2, . . . ,CL ⊆V,(∀v ∈V,(v ∈C1∨ v ∈C2∨·· ·∨ v ∈CL)) ∧(∀v ∈V, path_0(C1,v)∨ path_1(C1,v)∨·· ·∨ path_k(C1,v)) ∨(∀v ∈V, path_0(C2,v)∨ path_1(C2,v)∨·· ·∨ path_k(C2,v)) ∨. . . ∨

50

. . . ∨(∀v ∈V, path_0(CL,v)∨ path_1(CL,v)∨·· ·∨ path_k(CL,v)).

Now, every predicate path_i(C j,v) ∀i∈ {0, . . . ,k},∀ j ∈ {1,2, . . . ,L} has to be expressed in MSOLfollowing the constraints of k−path L−coloring. As in this chapter we will present the dynamicprogramming for the two main cases when k = 1 and k = 2, we develop the MSOL formulas forpath_0(C j,v), path_1(C j,v) and path_2(C j,v). The logic used in path_0 and path_1 are particularbut the logic used in path_2 could readily be extended to longer paths. Let edge(u,v) ≡ ∃(u,v) ∈ Eand edgeF(u,v)≡ ∃(u,v) ∈ F .

— path_0(C j,v): The vertex v is in color set C j and path of length 0 if there is no vertex u ∈C j

adjacent to V in G i.e. path_0(C j,v)≡ ¬∃u ∈C j,edge(u,v).— path_1(C j,v): The vertex v is in color set C j and path of length 1 if there is exactly one

vertex u ∈ C j such that (u,v) ∈ F . In other words, the vertex v has exactly one neighboru ∈ C j such that (u,v) ∈ F and u has no other neighbor than v in C j i.e. path_1(C j,v) ≡∃u ∈C j,edgeF(u,v)) ∧(¬∃u1 ∈C j,(u1 6= u 6= v)∧ (edge(u1,v))∧ (¬∃u2 ∈C j,(u2 6= u1 6= u 6=v)∧ (edge(u2,u)).

— path_2(C j,v): The vertex v is in color set C j and path of length 2 then v is either in the middleof the path then there should be u1,u2 ∈ C j such that (v,u1) ∈ F and (v,u2) ∈ F or v is anextremity of the path then there should be u1,u2 ∈C j such that (v,u1) ∈ F and (u1,u2) ∈ F . Inaddition, we have to verify that there are no other vertices (different from u1,u2) in C j adjacentto v,u1,u2. The corresponding MSOL formula: path_2(C j,v)≡ ∃u1,u2 ∈C j,(u1 6= u2 6= v)∧((edgeF(v,u1)∧ edge(v,u2))∨ (edgeF(v,u1)∧ edgeF(u1,u2)))∧(¬∃u3 ∈C j,(u3 6= u2 6= u1 6= v∧ edge(v,u3))∧(¬∃u4 ∈C j,(u4 6= u2 6= u1 6= v∧ edge(u1,u4))∧(¬∃u5 ∈C j,(u5 6= u2 6= u1 6= v∧ edge(u2,u5)).

3.3 Dynamic programming for proper 3-colorability

In this section, we present the dynamic programming approach applied to the proper 3-coloringproblem on graphs with bounded treewidth. First, we present the detailed steps of dynamic pro-gramming on a rooted tree decomposition, and give some illustrations based on the simple examplepresented in [53], see Fig. 3.4. As it is easier to run dynamic programming with a nice tree decom-position, we present later how to compute the tables corresponding to the four bags leaf, introduce,forget, join in the case of proper 3-coloring.

In dynamic programming for proper 3-coloring, we compute for every bag Xi ∈ X , a table notedA[i]. Every line of the table A[i] represents a potential solution (restricted to the nodes in Xi) and it isnoted A[i,ci] where i is the index of the corresponding bag and ci = [h1, . . . ,h|Xi|] is a vector of colorsassigned to the nodes in Xi. Every line A[i,ci] has a boolean value:

A[i,ci] =

{1 if ci can be extended to a proper 3−coloring of Vi.0 otherwise

In the computed tables, we save only the partial solutions that are valid ( A[i,ci] = 1 ).

3.3.1 Dynamic programming using tree decomposition

In the following, we explain the dynamic programming steps using a rooted tree decompositionon the example given in Fig. 3.4. Let G = (V,E) be the graph with V = {a,b,c,d,e} and E = {(a,b),

51

{b, c, d}

{a, b, c} {d, e}

Xr

X0 X1

(a) (b)

(c)

A[1, (red, green)] = 1

A[r, (red, green, blue)] = 1

i a b c0 red green blue1 red blue green2 green red blue3 green blue red4 blue red green5 blue green red

i b c d j0 red green blue {(4, 4), (4, 5)}1 red blue green {(2, 2), (2, 3)}2 green red blue {(5, 4), (5, 5)}3 green blue red {(0, 0), (0, 1)}4 blue red green {(3, 2), (3, 3)}5 blue green red {(1, 0), (1, 1)}

i d e0 red green1 red blue2 green red3 green blue4 blue red5 blue green

b

b

b

b ba

b

c

d e

A[0] A[1]

A[r]

Figure 3.4 – Dynamic programming tables example for proper 3-coloring problem using tree decom-position. (a) is an example of a graph G = (V,E). (b) is a tree decomposition (X ,T ) of the graph G.(c) present tables of dynamic programming for every bag of the tree decomposition T . Note that thecolumn j indicates how to complete the solution in the children bags.

52

(a,c),(b,c),(b,d),(c,d),(d,e)} (see Fig. 3.4(a)). Let (X ,−→T ) be the rooted tree decomposition of

G given in Fig. 3.4 (b): X = {Xr,X0,X1} where Xr = {b,c,d}, X0 = {a,b,c} and X1 = {d,e}. Thecomputed tables of each bag of the tree decomposition are presented in Fig. 3.4 (c). As we arelooking for a proper 3-coloring of the graph G, three colors are considered such that ∀ j ∈ {1, . . . |Xi|},h j ∈ {blue,green,red}.

— First, the tables corresponding to the leaf bags are computed: Assign a color to each vertex inthe leaf bag and verify that two vertices in the same bag that are adjacent in G receive differentcolors. In the example (c) in Fig. 3.4, the bag X1 has A[1,(red,green)] = 1 as setting colorred to d and color green to e is a proper 3-coloring of the vertices d,e.

— Second, the tables corresponding to the intermediate bags are computed:— Select an intermediate bag Xi, the computation of the table of Xi is done if and only if the

tables corresponding to its children are all computed.— Select a combination of partial solutions of the children bags that yields a (partial) proper

3-coloring of the vertices in Xi. In particular, if there are common vertices in the childrenbags they should have the same color in every partial solution of the selected combination.

— Assign a color to each vertex that appear in Xi without appearing in the children bags. Ifthe selected coloring of Xi is a proper 3-coloring then we save it in the table of Xi otherwisethe solution is not valid and we do not need to save it in the table.

In the example (c) in Fig. 3.4, the partial solution in line 0 in Xr’s table is either the extensionof the combination of the two partial solutions in line 4 in the table A[0] and in line 4 in thetable A[1]; or the extension of the combination of the partial solutions in line 4 in the table A[0]and in line 5 in the table A[1].

— Finally, the table corresponding to the root bag gives all the proper 3-coloring for the wholegraph G. In the example (c) in Fig. 3.4, all the partial solutions listed in the table of Xr, jointlywith the selected partial solutions of its children tables, allows to build all proper 3-coloringsolutions of the graph G.

Note: A partial solution in the table of an intermediate bag Xi can be extended to a proper 3-coloring of the subgraph induced by Vi: This statement is true because of the condition (3) in the treedecomposition definition. Assume for the sake of simplicity that Xi has only 2 children X j1 and X j2 .Let v1 be a vertex that appear in the subtree rooted at X j1 and let v2 be a vertex that appear in thesubtree rooted at X j2 and v1,v2 6∈ Xi. Therefore, there is no edge linking v1 and v2 in G otherwise v1and v2 have to appear in X j1 ,X j2 and Xi. Hence, if a combination of partial solutions in the childrentables of Xi in addition to the assignment of colors to the vertices that appear only in the bag Xi givesa partial solution of Xi∪X j1 ∪X j2 , it can be extended to solution for the subtree rooted at Xi.

3.3.2 Dynamic programming using nice tree decomposition

In this section, we present the computation of tables using a nice tree decomposition for proper3-coloring. It is enough to show how to extend partial solutions for each bag type (leaf, introduce,forget and join):

— Leaf bag: let Xi be a leaf bag, then Xi = {v} where v ∈V . The table A[i] is trivial. The node vcan receive either color blue or green or red. Therefore, there will be three lines in table A[i]:A[i, [blue]], A[i, [green]], A[i, [red]].

— Introduce bag: if Xi is an introduce bag and X j its child bag i.e. Xi = X j ∪{v}. Two casesmay appear:— The vertex v has no neighbor u in X j then any valid partial solution of X j is extended to a

valid partial solution of Xi for any color assigned to v.— There exists a vertex u ∈ X j such that u and v are adjacent in G then the selected valid

partial solution of X j is extended to a valid partial solution of Xi if and only if the color

53

b

vzb

b

w

(a) Xi bag and (u, v) ∈ F

u vb b

mi[u] = 1, ri[u] = 0mi[v] = 1, ri[v] = 0

u vb b

mi[u] = 0, ri[u] = 1mi[v] = 0, ri[v] = 0

b

b b

mi[z] = 1, ri[z] = 0

mi[v] = 0, ri[v] = 1mi[w] = 1, ri[w] = 0

y y

(b) Xi bag and (u, y) ∈ F (c) Xi bag and (v, y), (w, z) ∈ F

Figure 3.5 – Examples of the m and r values assigned to nodes in bags, given a snippet of a treedecomposition (X ,T ) of a graph G = (V,E) and F ⊆ E.

assigned to v is different from the color of all neighbors u.— Forget bag: if Xi is a forget bag and X j is its child bag i.e. Xi = X j\{v}. Every valid partial

solution of the bag X j can be extended to a valid partial solution for Xi, the color value of thevertex v is deleted.

— Join bag: if X j is a join bag where X j1 and X j2 are its two children i.e. X j1 = X j2 = Xi. Apartial solution of the table A[i] is valid if the same partial solution is valid for A[ j1] and validfor A[ j2].

3.4 Dynamic programming for 1-path 3-coloring problem

In this section, we present the dynamic programming approach for the 1-path coloring problemgiven a rooted nice tree decomposition with a small treewidth. Let us consider a graph G = (V,E),a set of edges F ⊆ E, the extremities of edges in F can receive the same color, a rooted nice treedecomposition (X ,

−→T ) of G and a set of 3 colors {red,blue,green}. The computation of the tables for

the four types of bags leaf, introduce, forget and join is inspired from the one presented in the case ofthe proper 3-coloring on a nice tree decomposition. The idea is to compute the tables correspondingto the bags in a bottom up way.

For every bag Xi ∈ X we define a table A[i]. Every line in the table A[i] represent a potential partialsolution restricted to vertices in Xi. It is of the form A[i,ci,mi,ri] where i is the index of the bag Xi, ci

is the vector of the colors assigned to nodes in Xi, mi is a boolean vector that indicates if a vertex v∈ Xi

has a neighbor u ∈ Xi with the same color and (u,v) ∈ F , and ri is a boolean vector that indicates ifa vertex v ∈ Xi has a neighbor outside Xi, i.e. u ∈ Vi \Xi, with the same color and (u,v) ∈ F . The mi

and ri vectors are important to construct the 1-path coloring solution, see Fig. 3.5. A[i,ci,mi,ri] is aboolean value (and again we need to keep only solutions that are valid).

A[i,ci,mi,ri] =

{1 if ci can be extended to a 1-path 3-coloring of Vi,0 otherwise.

The computation of tables for the bags leaf, introduce, forget and join are presented in the follow-ing:

— Leaf bag: let Xi be a leaf bag where Xi = {v}. Every line A[i,ci,mi,ri] is a valid partial solutionof Xi for every value of ci(v) ∈ {red,blue,green}, mi(v) = 0 and ri(v) = 0 (Xi does not haveany child bag and v does not have a neighbor in Xi and outside Xi yet).

— Introduce bag: let Xi be an introduce bag and X j its child bag and v is the introduced vertexto X j i.e. Xi = X j ∪{v}. Two cases may arise:

54

vXi

Xjw

bbw

b . . .

. . .v Xi

Xjw

b w

b

. . .

. . . v Xi

Xjw

bb w

b . . .

. . .

b

b

vXi

Xjw

bbw

b . . .

. . . vXi

Xjw

bbw

b . . .

. . . vXi

Xjw

b

bw

b . . .

. . .

bz

bz

bb b

bbz

z

zb

zb

zb

Figure 3.6 – Illustration of the valid and non valid extension of the partial solutions for the introducebag, in the case of 1-path coloring problem.

— First, if v has no neighbor u ∈ X j then every valid partial solution in the table A[ j] isextended to a valid partial solution in the table A[i] for any assigned color to v. Of course,it is natural to set mi(v) = ri(v) = 0, because v has no neighbor inside and outside Xi yet.

— Second, if v has at least one neighbor in X j, then a valid partial solution in the table A[ j] isextended to a valid partial solution in the table A[i] if an only if the two following conditionsare verified: v receives the same color as at most one neighbor u ∈ X j such that (u,v) ∈ Fand u has no neighbor inside and outside X j with the same color (i.e. m j(u) = r j(u) = 0),otherwise a path of length > 1 will be created; and v receives different colors than each ofits neighbors w ∈ Xi such that (v,w) ∈ E \F .

See Fig. 3.6 for an illustration of the listed cases.— Forget bag: let Xi be a forget bag and X j its child bag and v the forgotten vertex i.e. Xi =

X j\{v}. Partial solutions in the table A[ j] are extended to partial solutions in the table A[i] byremoving the values corresponding to the vertex v from the vectors c j,m j and r j.

— Join bag: let Xi be a join bag. Let X j1 and X j2 be its two children bags where X j1 = X j2 = Xi.A partial solution in the table A[i] is valid if the same partial solution is valid for A[ j1] andfor A[ j2] except: if a vertex v in the solution of A[ j1] has same color as one of its neighboru outside of X j1 such that (u,v) ∈ F and in the solution of A[ j2] the vertex v has same coloras one on its neighbors w outside of X j2 such that (v,w) ∈ F , the combination of these twosolutions generate a solution in A[i] where v has two neighbors with the same color, it createsa path of length 2 and we are looking for path of length at most 1. Examples of valid and nonvalid combinations are presented in Fig. 3.7.

3.5 Dynamic programming for induced 2-path 3-coloring problem

In this section, we present the dynamic programming approach in the case of induced 2-path col-oring using a nice tree decomposition with a small treewidth. The computation of tables of the nicetree decomposition is inspired from the computation of tables in the case of 1−path coloring problem,as described in the previous section.

55

v Xi

Xj1

bw

bu

b

wb

bu

bv

Xj2

wb

bu

bv

v Xi

Xj1

bw

bu

b

wb

bu

bv

Xj2

wb

bu

bv

v Xi

Xj1

bw

bu

b

wb

bu

bv

Xj2

wb

bu

bv

rj1(u) = 1 rj2(u) = 0 rj1(u) = 0 rj2(u) = 0

rj1(u) = 0 rj2(u) = 1

b

b

Xj1

wb

bu

bv

Xj2

wb

bu

bv

v Xibw

bu

b

b b

b b

xy

x y

rj1(u) = 1 rj2(u) = 1

v Xi

Xj1

bw

bu

b

wb

bu

bv

Xj2

wb

bu

bv

rj1(u) = 0 rj2(u) = 1

b

b

Figure 3.7 – Allowed and forbidden extensions of partial solutions for the join bag, in the case of1-path coloring problem.

For the 2-path coloring problem, every line of the table A[i], corresponding to a bag Xi, is a partialsolution of the vertices in Xi defined with A[i,ci, f 0

i , f 1i , f 2

i ] where i is the index of the correspondingbag Xi, ci is the vector of colors assigned to nodes in the bag Xi, f 0

i is a boolean vector assigned tovertices in Xi such that ∀v ∈ Xi, f 0

i (v) = 1 iff v has no neighbor with same color inside and outsideXi, f 1

i is a boolean vector assigned to vertices in Xi such that ∀v ∈ Xi, f 1i (v) = 1 iff v has 1 neighbor

u ∈Vi with the same color where (u,v) ∈ F , f 2i is a boolean vector assigned to vertices in Xi such that

∀v ∈ Xi, f 2i (v) = 1 iff v is part of a path of length 2 where all its edges are in F , all its vertices are in Vi

and have the same color. Thus, ∀v ∈ Xi, f 0i (v)+ f 1

i (v)+ f 2i (v) = 1. In other words, for table A[i], each

line A[i,ci, f 0i , f 1

i , f 2i ] is a boolean value defined as follows:

A[i,ci, f 0i , f 1

i , f 2i ] =

{1 if ci can be extended to a 2-path 3-coloring of Vi,0 otherwise

The computation of tables for the leaf, introduce, forget and join bags are presented in the follow-ing:

— Leaf bag: let Xi be a leaf bag, Xi = {v}. As Xi does not have any child bag and v is the singlevertex in Xi, every valid partial solution A[i,ci, f 0

i , f 1i , f 2

i ] = 1 is defined for every value ofci(v) ∈ {green,red,blue}, f 0

i (v) = 1, f 1i (v) = 0 and f 2

i (v) = 0 , otherwise A[i,ci, f 0i , f 1

i , f 2i ] =

0.— Introduce bag: let Xi be an introduce bag where Xi = X j ∪{v}. Two situations are possible:

— First, if v has no neighbor u ∈ X j then every valid partial solution A[ j,c j, f 0j , f 1

j , f 2j ] = 1 is

extended to a valid partial solution in A[i] for every color assigned to v and fixing f 0i (v) = 1,

f 1i (v) = 0, f 2

i (v) = 0.— Second, if v has neighbors in X j then every partial solution A[ j,c j, f 0

j , f 1j , f 2

j ] = 1 is ex-tended to a valid partial solution in the table A[i] such that ∀u ∈ X j where (u,v) ∈ E \F ,

56

u and v receive different colors i.e. ci(v) 6= c j(u), otherwise the color assigned to v can bethe same as:— one neighbor u∈X j where (u,v)∈F such that u has no neighbor with the same color in

A[ j,c j, f 0j , f 1

j , f 2j ] i.e f 0

j (u) = 1, then ci(v) = c j(u) and f 1i (v) = f 1

i (u) = 1. Or u has ex-actly one neighbor w (that is not adjacent to v) with the same color in A[ j,c j, f 0

j , f 1j , f 2

j ]

i.e. f 1j (u) = 1 and f 1

j (w) = 1. Then, the extended solution updates the following valuesci(v) = c j(u) = c j(w) and f 2

i (v) = f 2i (u) = f 2

i (w) = 1.— exactly two neighbors u,w∈ X j having the same color where (u,v),(v,w)∈ F , (u,w) 6∈

E and every node u and w has no neighbor with the same color in the selected partialsolution A[ j,c j, f 0

j , f 1j , f 2

j ] i.e. f 0j (u) = 1 and f 0

j (w) = 1. Then the extended solutionupdates the following values ci(v) = c j(u) = c j(w) and f 2

i (v) = f 2i (u) = f 2

i (w) = 1.— All the other configurations are not valid because they generate a k−path coloring with

k > 2. See figures 3.8 for illustrations of the extension of partial solutions in the introducebag case.

— Forget bag: let Xi be a forget bag, Xi = X j\{v}. A valid partial solution A[ j,c j, f 0j , f 1

j , f 2j ] = 1

is extended to a valid partial solution of the table A[i] by removing the values corresponding tothe forgotten vertex v from the vectors c j, f 0

j , f 1j , f 2

j .— Join bag: let Xi be a join bag where Xi = X j1 = X j2 . A partial solution in the table A[i] is valid

if the same partial solution is valid for A[ j1] and for A[ j2] verifying the following constraints:— The two selected vector of colors in the partial solutions from A[ j1] and A[ j2] are equal i.e.

c j1 = c j2 .— Each vertex v∈ X j1 having f 2

j1(v) = 1 in the solution of A[ j1], and f 2j2(v) = 1 in the solution

of A[ j2], v is in a path p of length 2 where all its vertices have the same color and the edgesof the path are in F . Then all the nodes within the path p have to be in the bag X j1 (hencein X j2 and Xi).

— Each vertex v ∈ X j1 with f 2j1(v) = 1 in the solution of A[ j1], and f 1

j1(v) = 1 in the solutionof A[ j2]. Then v must have at least one neighbor with the same color in the bag X j1 (hencein X j2 and Xi).

If the listed conditions are not verified then the connected components in each color will notform a path of length at most 2. See Fig. 3.9 for illustrations of valid and non valid join bagcases.

3.6 Benchmark instances

In this study, we use 23 industrial instances, 10 of which are the one presented in the ILP formu-lations chapter and the remainder are other industrial instances of via layouts from Mentor Graphics(that we obtained later). They are displayed in Table 3.1: « Graph » gives the name of the graph,« |V | » is the number of nodes, « |E| » is the number of edges, « |F | » is the number of edges that canbe assigned to a path, « Density » is the density of the graph that is computed as follows: 2|E|

|V |(|V |−1) ,« ω(G) » is the size of the maximum clique of the graph, « ∆(G) » is the maximum vertex degree inthe graph, « Width » is the width of the tree decomposition corresponding to the graph, we calculatedit as the minimum width over 10 iterations of a heuristic computing a tree decomposition in D-FLATframework (see next section for a presentation of the D-FLAT framework). Note that we did not usethe grid graph instances as presented in the previous chapter, because it is known that the completegrid graphs of size n×n have a treewidth equal to n. After several random generation of grid graphsG = (V,E) where the probability to choose a vertex in the grid as a vertex of the graph is p = 0.5 wefound that the treewidth of the grid graphs is very high, for instance a graph having 1900 nodes canhave a width of 40, which makes the dynamic programming unpractical on these data.

57

vXi

Xjw

bbw

b . . .

. . .v Xi

Xjw

b

b . . .

. . .

Xi

Xjwb . . .

Xi

Xjwb . . .

b

b

z

z

zb

b

bu

u

Xi

Xjwb . . . b

z

bu

vXi

Xjw

bbw

b . . .

. . .v

Xi

Xjw

bbw

b . . .

. . . vXi

Xjw

bbw

b . . .

. . .

bz

vXi

Xjw

bbw

b . . .

. . .Xi

Xjwb . . .b

zb

b

b

bu

vXi

Xjw

bbw

b . . .

. . .

bz

bu

b

b b

b

b

b

b

. . .z

b

z

z

z

b

bu

u

z

b

b

u

z

v b

wb . . .b

zbu

w. . .z

u

b vb

b

b w. . .zu

b vbbb w

. . .

z

u

b vb

b

b

w . . . zbb

Figure 3.8 – Illustrations of valid and non valid extension of partial solutions for the introduce bag(Xi = X j ∪{v}), in the case of 2-path coloring.

58

Xj1

wb

bu

bv

Xj2

wb

bu

bv

v Xibw

bu

b

b b

b b

x y

x y

f2j1(u) = 1 f2

j2(u) = 1

ba

bz

bz

ba

Xj1

wb

bu

bv

Xj2

wb

bu

bv

v Xibw

bu

b

b b

b

x y

x

f2j1(u) = 1 f2

j2(u) = 1

by

Xj1

wb

bu

bv

Xj2

wb

bu

bv

v Xibw

bu

b

b b

b

x z

xby

by

ba

bab

z

Xj1

wb

bu

bv

Xj2

wb

bu

bv

v Xibw

bu

b

b b

b

x y

x

f2j1(u) = 1 f2

j2(u) = 1

by

f2j1(u) = 1 f2

j2(u) = 1

v Xi

Xj1

bw

bu

b

wb

bu

bv

Xj2

wb

bu

bv

v Xi

Xj1

bw

bu

b

wb

bu

bv

Xj2

wb

bu

bv

v Xi

Xj1

bw

bu

b

wb

bu

bv

Xj2

wb

bu

bv

b

b

Xj1

wb

bu

bv

Xj2

wb

bu

bv

v Xibw

bu

b

b b

b b

xy

x y

Xj1

wb

bu

bv

Xj2

wb

bu

bv

v Xibw

bu

b

b

b b

x

x y

by

Xj1

wb

bu

bv

Xj2

wb

bu

bv

v Xibw

bu

b

by

by

Xj1

wb

bu

bv

Xj2

wb

bu

bv

v Xibw

bu

b

b b

b

x y

x

f2j1(u) = 1 f1

j2(u) = 1

by

Figure 3.9 – Valid and non valid configurations for the extension of partial solutions in a join bagXi = X j1 ∪X j2 , in case of induced 2-path coloring.

59

Graph |V | |E| |F | Density ω(G) ∆(G) Widthnew_clip_1 54 56 52 0.0391 3 4 2new_clip_2 61 86 85 0.0469 3 5 2new_clip_3 89 112 107 0.0286 3 5 2new_clip_4 90 113 109 0.0282 3 5 2new_clip_5 96 111 105 0.0243 3 4 2new_clip_6 98 126 120 0.265 3 5 3new_clip_7 102 144 141 0.0279 3 5 2new_clip_8 111 140 137 0.0229 3 5 2new_clip_9 114 131 125 0.0203 3 4 2new_clip_10 116 155 151 0.0232 3 5 3new_clip_11 119 142 136 0.0202 3 4 3new_clip_12 128 159 149 0.0195 3 5 2new_clip_13 137 167 160 0.0179 3 5 3new_clip_14 159 196 188 0.0156 3 5 2new_clip_15 173 224 216 0.0150 3 5 3new_clip_16 382 396 339 0.0054 3 4 2new_clip_17 969 1001 900 0.0021 3 3 2new_clip_18 993 1009 927 0.0020 3 4 2new_clip_19 997 1047 906 0.0021 3 4 2new_clip_20 998 1024 924 0.0020 3 4 2new_clip_21 1900 1937 1804 0.0010 3 4 2new_clip_22 1912 1960 1809 0.0010 3 4 2new_clip_23 1937 1996 1812 0.0010 3 4 2

Table 3.1 – Industrial instances for the dynamic programming technique.

60

Graph |V | χ1ind cpu time (sec) χ2

ind cpu time (sec)new_clip_1 54 2 0.35 2 0.92new_clip_2 61 2 0.75 2 1.62new_clip_3 89 2 0.51 2 1.5new_clip_4 90 2 0.98 2 1.21new_clip_5 96 2 0.55 2 1.73new_clip_6 98 2 0.84 2 3.45new_clip_7 102 2 0.94 2 1.47new_clip_8 111 2 1.45 2 2.21new_clip_9 114 2 0.79 2 2.21new_clip_10 116 2 1.15 2 2.5new_clip_11 119 2 1.41 2 4.02new_clip_12 128 2 1.17 2 2.83new_clip_13 137 2 1.22 2 3.05new_clip_14 159 2 1.42 2 3.53new_clip_15 173 2 1.83 2 3.44new_clip_16 382 2 2.57 2 4.95new_clip_17 969 2 7.26 2 12.52new_clip_18 993 2 6.19 2 12.63new_clip_19 997 3 8.86 2 12.82new_clip_20 998 3 8.65 2 12.95new_clip_21 1900 3 21.09 2 26.87new_clip_22 1912 3 18.22 2 26new_clip_23 1937 2 6.29 2 27.02

Table 3.2 – Results of dynamic programming for 1-path coloring and 2-path coloring on industrialinstances using D-FLAT framework.

3.7 Numerical results

The numerical results are presented in the case of induced k−path coloring problem where k = 1and k = 2. The dynamic programming conditions presented in Section 3.4 and Section 3.5 were imple-mented using D-FLAT 1.2 framework. D-FLAT [53] is a framework that already implements a genericversion of the dynamic programming algorithm: it uses a heuristic to calculate a tree decompositionof a given graph, then a logic laguage called Answer Set Programming (ASP) is used as an interface todescribe the tables generations of our problem. A manual of D-FLAT can be found in [53] and moredetails about the ASP laguage can be found in [16]. Moreover, see Appendix C for the D-FLAT codeused for the induced k−path coloring problem when k = 1 and k = 2. All the tests were done on a ma-chine equipped with an Intel(R) Xeon(R) CPU E5-2640 2.60 GHz and a memory of 529GB. D-FLATreturned a coloring solution for a fixed set of colors, we first run all the instances using two colors, ifa solution was returned by D-FLAT then we saved it and saved the computation time, if no solutionis found then we increased the set of colors by one until a solution was found. The Table 3.2 presentsthe results of the minimum number of colors in the case of 1−path coloring in column « χ1

ind », in thecase of 2−path coloring in column « χ2

ind » and their corresponding CPU time in column « cpu time(sec) » for every industrial instance.

The results presented in Table 3.2 show that the dynamic programming approach using D-FLATframework return very good results on the industrial instances, optimal solutions are found in fewseconds. Note that the instances presented in this study are the largest connected components of every

61

vias layout. We can conclude that dynamic programming is a promising method to solve the wholelayout designs by applying the same process on all the connected components of a given vias layoutin a very short runtime, few minutes. This result can be expected for the used benchmark instancesbecause we know that each vias layout contain a hundred of connected components, we can assumethe runtime of the dynamic programming on the connected components will be less than the one spenton the largest one.

3.8 Conclusion

The dynamic programming results are comparable to the one of the naïve and IAC formulationspresented in Chapter 2, Section 2.3.1 on industrial instances. Note that the results work pretty wellbecause the treewidth is very small on the 23 industrial instances (at most 3). It is difficult to guarantythrough that, all industrial instances will have this property.

62

Chapter 4

k-matching cover

In this chapter we present an alternative model to solve the DSA-aware MP problem. We focus ona special case which reduces to the so called k−matching cover problem, and we thus study the corre-sponding problem. When k = 1, the k−matching cover problem is equivalent to the perfect matchingproblem and is thus polynomially solvable [52]. When k≥ 2, the problem is still polynomial [67], andsomewhat simpler, because it can be modeled as a union of k matching matroids and it can thus besolved efficiently through matroid intersection (see for instance [64]). In [67], Wang et al proposed analgorithm for this latter case based on Edmonds-Gallai Theorem and several technical augmentationalgorithms. Their proof contained some flaws though. Inspired by their work, we developed a newflawless algorithm for the problem that somehow exploit again Edmonds-Gallai decomposition resultsin conjunction with simple flow arguments. In the mean time, Wang et al corrected their flaws [68].We then developed a much simpler algorithm based on bipartite matching. Before presenting thesetwo results we present key ingredients from matching in bipartite and non bipartite graphs and networkflows that are needed in our proofs.

4.1 Notations and definitions

Let G = (V,E) be a graph. Note that we focus only on simple graphs i.e. graphs without loopsand parallel edges. In this chapter we use the standard graph theory notations and definitions that arereminded in the Appendix A. Other main definitions used in this chapter are presented below.

A matching M of G is a subset of edges of E where no two edges in M share the same vertex. Thesize of a matching M is the cardinality of the set M. M is maximum if it is of maximum size over allmatchings of G. We denote the size of a maximum matching by ν(G). Given a matching M, a vertexv ∈V is said to be exposed if it is not incident to any edge e ∈M. M is perfect if each vertex u ∈V isincident to an edge in M, an example is presented in Fig. 4.1 (a). A stable set is a set of pairwise non-adjacent vertices. The maximum matching problem is the problem of finding a matching of maximumcardinality in a graph. The maximum weight matching problem is the problem of finding a matchingwhere the sum of the edges weight in the matching is maximum. The maximum (weight) matchingproblem is known to be polynomial. In the case of bipartite graphs, the cardinality version reduces toa maximum flow problem (see Section 4.3.2 for a proper defition). Moreover, the Hungarian method,developed by Kuhn [50], allows to find a maximum weight matching in polynomial time. In thecase of general graphs, Edmonds’ algorithm finds a maximum (weighted) matching in polynomialtime [13]. The matching cover problem is the problem of finding a minimum number of matchingswhose union covers all the vertices of a given graph. Obviously the problem reduces to the questionof whether a graph G has a matching cover of size k for all k = 1, ..., |E|. A formal definition of thek−matching cover is given in the following:

63

Definition 4.1. (k−matching cover). Let G = (V,E) be a graph. Let k≥ 1 be an integer, a k-matchingcover of G is a set of k (disjoint) matchings M1, . . . ,Mk of G where each vertex of V is incident to atleast one edge in ∪k

i=1Mi.

A minimum matching cover is a k-matching cover with k minimum. See Fig. 4.1 for some exam-ples of k−matching cover.

b b

b bb

bb

b

b b

bb

(a) (b) (c)

bb

Figure 4.1 – Examples of k-matching cover. (a) a 1-matching cover a.k.a a perfect matching. (b) a2-matching cover. (c) a 4-matching cover. (Each color represents a matching).

4.2 From DSA-aware MP to k−matching cover

In this section, we present the connection between the DSA-aware MP problem and the k−matchingcover problem. Remember that the DSA-aware MP problem can be defined as a general or inducedk−path coloring problem for a graph G= (V,E), F ⊆ E and k integer. As presented in Chapter 2, thereis a naïve formulation to solve the k−path coloring that consist in listing all feasible paths of length atmost k (including singletons). Let us call the vertices within each feasible path of length at most k agroup. The model selects among those groups some that covers all the vertices of the original graphand simultaneously minimizes the chromatic number of the graph induced by these groups. Moreformally, we can create an auxiliary graph G′ = (V ′,E ′) such that every vertex of V ′ is a listed group.Two vertices are adjacent in G′ if there exists an edge in E(G) between the corresponding groups or ifthe two groups contain a same vertex. An example of an auxiliary graph is presented in Fig. 4.2(b).

b

b b b b

b b

Edges in F :

Edges in E\F :

(a) G = (V,E) and F ⊆ E,

b

b b b b

b b

b

b

b

ba b

c

d e

fgc fg

a b d e

{a, b}

{g, d}

{e, f}

(b) The auxiliary graph G′ = (V ′, E′) of G.

Figure 4.2 – An example of a graph G and its auxiliary graph G′. (a) presents a graph G modeling a lay-out of vias where V = {a,b,c,d,e, f ,g} and E = {(a,b),(a,c),(b,c),(b,d),(d,e),(d,g),(e, f ),( f ,g)}and F = {(a,b),(d,g),(e, f )} and consider a 2-path coloring of G. (b) presents the auxiliary graph G′

of G where the listed groups are: {a,b,c,d,e, f ,g,{a,b},{g,d},{e, f}}.

Now we want to find a set of vertices U in G′ such that (i) the union of the groups correspond-ing to the vertices in U covers all original vertices of G and (ii) the chromatic number of G′[U ] is

64

minimum among all such U . For instance in Fig. 4.2(b) the set U can have the following valueU := {{a,b},c,{g,d},{e, f}} and χ(G′[U ]) = 2. Observe that the set of vertices in G′ containing agiven vertex in G forms a clique (non necessarily maximal), examples of such cliques can be seen inFig. 4.2(b), for instance {a,{a,b}} or {d,{g,d}}. Let us call K the set of all such cliques. Findinga minimum k−path coloring of G is equivalent to determine the minimum number of colors neededto partially color the vertices of G′ so that each clique of K is hit by (at least) one of the colors. Wemight call the underlying graph coloring problem the clique hitting partial coloring problem. For-mal definitions are given in Definition 4.2 and Definition 4.3. See Fig. 4.3 for a clique hitting partialcoloring illustration.

Definition 4.2. (Partial coloring). Let G= (V,E) be a graph. A partial coloring of G is an assignmentof colors to a subset of vertices of V such that no two adjacent vertices receive the same color. Whenthe number of colors used is k we call it a partial k−coloring of G.

Definition 4.3. (K −clique hitting partial coloring). Let G = (V,E) be a graph. Let K be a subsetof cliques of G. A K −clique hitting partial coloring of G is a partial coloring where in each K , atleast one vertex receives a color. When the number of colors used is set to k we call it a K −cliquehitting partial k−coloring of G.

b

b b b b

b b

b

b

b

c fg

a b d e

{a, b}

{g, d}

{e, f}

(c) Cliques in K are selected in bold grey.

b b

b b b b

b b

b

b

b

b

c fg

a b d e

{a, b}

{g, d}

{e, f}

(d) A clique hitting partial coloring of G′

b

b

b

b

b

b

Figure 4.3 – Example of clique hitting partial coloring.

Given a set of cliques K , the K −clique hitting partial coloring problem is the problem of findingthe minimum number of colors k such that a K −clique hitting partial k−coloring exists. An exampleis presented in Fig. 4.3 where figure (c) shows the set K corresponding to the auxiliary graph G′ pre-sented in Fig. 4.2, and figure (d) presents a K −clique hitting partial coloring of G′ using two colors.The K −clique hitting partial coloring problem is obviously a hard problem as it contains (proper)graph coloring as a special case: for a given graph, G = (V,E) we can choose each single vertex asone of the cliques of K i.e K = {{v},v ∈V}, an example is presented in Fig. 4.4. Furthermore, thek-matching cover appears to be a special case of the K −clique hitting partial coloring problem if theauxiliary graph G′ is the line graph of a graph H, (V (G′) = E(H) and two vertices are adjacent inG′ iff their corresponding edges in H are incident). In the definition of line graphs, one can see thata star in H yield a clique in G′. Therefore, if we choose K in G′ as the cliques stemming from themaximal stars in H incident to each vertex i.e. K = {{e ∈ δ (v)} : v ∈ V (H)}, finding a minimumnumber of colors for the K -clique hitting partial coloring of G′ is equivalent to find a minimum k-matching cover of H. Indeed, a matching M ⊆ E(H) is a stable set of G′. Therefore a k-matchingcover {M1, . . . ,Mk} is a set of k stable sets of G′ that color at least one vertex for each clique in K .

65

b

b b b b

b b

(e) G = (V,E), F ⊆ E, F = ∅

a b

c

d e

fg

b

b b b b

b b

a b

c

d e

fg

bbb

b

b

bbb

bbb

b

(f) K = {{v}, v ∈ V }

b

b b b b

b b

a b

c

d e

fg

bbb

b

b

bbb

bbb

bb

b b b b

bb

is a proper coloring of G

(g) A K−clique partial coloring of G

Figure 4.4 – A particular case of clique hitting partial coloring example. When K = {{v},v ∈ V},the K −clique hitting partial coloring of G is equivalent to a (proper) coloring of G.

4.3 Matching and network flow

In this section we present some standard theorems and algorithms for matching in bipartite andnon bipartite graphs as well as network flows. These results are key ingredients for the proof of thek-matching cover theorem in the next section.

4.3.1 Matching

The matching problem was first studied on bipartite graphs before being extended to generalgraphs. Let G = (X ∪Y,E) be a bipartite graph, Hall’s theorem [33] gives a characterization forthe existence of a perfect matching. It essentially states that there should be enough vertices in theneighborhood of any subset S of vertices of X or Y , see Theorem 4.1.

Theorem 4.1. (Hall’s theorem [33]). Let G = (X ∪Y,E) be a bipartite graph. Then G has a perfectmatching if and only if |S| ≤ |N(S)|, for each S⊆ X and S⊆ Y .

Hall’s necessary and sufficient condition for perfect matching can be extended to a necessarycondition for the existence of a perfect matching in general graphs, by requiring that |S| ≤ |N(S)|, forall stable sets S. Unfortunately, the condition is not sufficient for general graphs as illustrated by atriangle. Tutte [65] presented a characterization of the existence of a perfect matching in an arbitrarygraph. He stated that a graph G has a perfect matching if for every set of vertices U , the number ofodd components in G−U is at most the cardinality of U as presented in Theorem 4.2.

Theorem 4.2. (Tutte’s theorem [65]). Let G = (V,E) be a graph. There exists a perfect matching inG if and only if for every U ⊆V , o(G−U)≤ |U |, where o(G−U) is the number of odd componentsof G−U.

In [10], Berge generalized Tutte’s theorem and gave a characterization of the size of the maximummatching in an arbitrary graph as presented in Theorem 4.3.

Theorem 4.3. (Tutte-Berge formula [10]). Let G = (V,E) be a graph. The size of the maximum

cardinality matching in G denoted by ν(G) is given by: ν(G) =12

minU⊆V

(|U |−o(G−U)+ |V |), where

o(G−U) is the number of odd components of G−U.

The size of a maximum matching is always at most12

minU⊆V

(|U |−o(G−U)+ |V |) and there al-

ways exists a subset of vertices U that verify the equality given by the Tutte-Berge formula, this setis called a Tutte set. Such a set U can be found at the end of Edmonds algorithm: it is the set of evenvertices (see Edmonds algorithm below). We can see that Tutte’s theorem is a particular case of theTutte-Berge formula.

66

In the case of bipartite graphs, we can extend Hall’s Theorem from perfect matching to k−matchingcover as presented in Theorem 4.4. Quite surprisingly, this result holds for general graphs if we slightlyadapt the theorem as long as k≥ 2, as presented in Theorem 4.5. We will prove this more general ver-sion in the next section, Theorem4.4 will follow then as a simple corollary.

Theorem 4.4. Let G = (X ∪Y,E) be a bipartite graph and let k ∈ N\{0}. There exists a k-matchingcover of G if and only if ∀S⊆ X and ∀S⊆ Y , |S| ≤ |N(S)| · k.

Theorem 4.5. Let G = (V,E) be a graph. Let k ≥ 2 be an integer. There exists a k-matching cover ofG if and only if for all stable set S of G, |S| ≤ |N(S)| · k.

Let us now recall some algorithmic aspects of the matching problem. Most algorithms for bipartiteand non-bipartite graphs are built upon Berge’s augmenting path theorem [11]. Before introducing theBerge’s theorem we define the notion of M−alternating path and M−augmenting path. Illustrationsare presented in Fig. 4.5.

Definition 4.4. (M-alternating path). Let G = (V,E) be a graph. Let M be a matching of G. AM-alternating path P is a path of G where edges are alternatively in M and E \M (or vice-versa).

Definition 4.5. (M-augmenting path). Let G = (V,E) be a graph. Let M be a matching of G. An M-augmenting path P is an alternating path where the two extremities of P are (distinct) exposed verticesi.e. the edges incident to the extremities of the path are in E \M.

b b b b b b

b b b b b b

(a) Alternating path

(b) Augmenting path

exposed vertex

Matching edges

Figure 4.5 – An alternating path, an augmenting path and an exposed vertex.

Theorem 4.6. (Berge’s theorem [11]). Let G = (V,E) be a graph. A matching M of a graph G ismaximum if and only if there is no M-augmenting path in G.

Berge’s theorem reduces the (maximum size) matching problem to a sequence of augmentationproblems: one can start with an initial matching M and check whether an M-augmenting path exists;if so, one can produce a matching of size |M|+ 1 and iterates with the new matching until no M-augmenting path is found.

The detection of an augmenting path can somewhat be reduced to the detection of a regular pathbetween exposed vertices in an auxiliary graph H that is built as follows: The vertices of H are theoriginal vertices of G and an arc (u,v) exists in H if and only if there exists w ∈V where w 6= u, w 6= vsuch that (u,w) ∈ E and (w,v) ∈M. It can be easily shown that, to a M-augmenting path with extrem-ities u,v in G corresponds a path in H with extremities u and a vertex incident to v in G. Similarly, toa path in H between an exposed vertex u and a vertex incident to an exposed vertex v corresponds anM-augmenting walk (a walk W in contrast with a path that might visit several times a same vertex) inG with extremities u and v. Now when the graph is bipartite this walk is actually a M-augmenting pathin G and we thus have a simple algorithm (the reader can refer to [52] for more details on the bipartite

67

case).

In contrast, in general graphs, the presence of odd-cycles destoys this one-to-one correspondencebetween M-alternating paths of G and their natural counterpart in H. Edmonds [24] solved the problemby shrinking the odd cycles into super vertices. We define a M-blossom B in G to be an odd cycle ofG (when M is clear from the context, we simply call B a blossom). Shrinking a blossom B into a supervertex v is the operation of replacing all vertices and edges within B by a vertex vB. Consequently, anyedge incident to some vertex in V (B) is now replaced by an edge incident to vB, as shown in Fig. 4.6.The resulting graph is denoted by G\B with the set of vertices V (G\B) :=V (G)\V (B)∪{vB} and theset of edges E(G\B) := {(u,v) ∈ E : u,v ∈V (G)\V (B)}∪{(u,vB) : u ∈V (G)\V (B) and (u,v) ∈ Efor some v ∈V (B)}. We denote by M \B the matching of G\B defined by M−{(u,v) : u,v ∈V (B)}.Lemma 4.1 [24] gives a direct relation between maximum matchings in G\B and maximum matchingsin G. Note that the proof of Lemma 4.1 is algorithmic and one can easily retrieve a maximum matchingin G from a maximum matching in G\B, see for instance [52].

b bbb b

bb

bb

b bbb b

(a)

(b)

vB

Blossom B

b

b

b

bb

b

Figure 4.6 – An example of a blossom in (a) and its shrinking to super vertex in (b).

Lemma 4.1. (Edmonds [24]). Let G = (V,E) be a graph, M a matching of G and B a blossom. M \Bis a maximum matching in G\B if and only if M is a maximum matching in G.

We will now describe this algorithm in details as presented in [13]. While we refer to textbooksfor the proofs of some standard results, we detail some here in order to exhibit a few key propertiesthat we will use later. Edmonds’ algorithm proceeds as follows: Let M be a matching of G and S⊆Vbe the set of exposed vertices. If S = /0 then G has a perfect matching and we have found a maxi-mum matching. If S 6= /0 then the algorithm attempts to detect a M-augmenting path by constructingiteratively a M-alternating forest. A M-alternating forest F is a forest of G where each componentof F contains exactly one exposed vertex called the root of the component, each exposed vertex of Sbelongs to exactly one component of F , and each edge (u,v) of F belongs to the matching M iff thevertex u is at an odd distance from the root and the vertex v is at an even distance from the root. Inparticular, each vertex in F which is at an odd distance from any vertex in S has degree 2 in F andthese vertices are called inner vertices, while the remaining vertices in F are called outer vertices. Bydefinition, the root vertices of the components of F are outer vertices. An alternating forest exampleis presented in Fig. 4.7.

For convenience, in the following algorithm description, given a vertex u with (u,v) ∈ M, wedenote v by M(u). Also for a vertex x, we denote by P(x) the unique alternating path from x to the

68

bc bc bcbc outer vertices

inner vertices

matching edges

bc

b

b b b b

b b b b b b

bc bc bc

bc bc bc bcbc bc

bc

Figure 4.7 – An alternating forest example.

set of exposed vertices S in the alternating forest F . Edmonds’ algorithm starts by initializing thealternating forest F with the vertices in S and no edges, and iterates according to the four situationslisted below:

1. It grows F: if there is a vertex y not in F and y is adjacent to an outer vertex x in F , then wegrow F by adding {(x,y),(y,M(y))}.

2. It detects a blossom B: if x,y are two outer vertices in a same component of F , which areadjacent in G, then P(x)∪ (x,y)∪P(y) is an alternating walk in G and there is a blossom B inP(x)∪ (x,y)∪P(y). We shrink B and we proceed in G \B with M \B following Lemma 4.1.Then we can use F \B as an initial forest.

3. It finds an augmenting path: if x,y are two outer vertices in two different components ofF , which are adjacent in G, then P(x)∪ (x,y)∪P(y) is an alternating path in G, then M isaugmented along this path (we extend it to an augmenting path of the original graph if weshrunk blossoms in previous iterations). The algorithm starts again with the new matchingusing Berge’s theorem, see Theorem 4.6.

4. It stops: if none of the above applies, we stop as M is a maximum matching (the algorithmexhibits a certificate of optimality i.e. a Tutte set).

b b

b b

b b

b b

b b

b

b

b

b

b

b

1

2

3

4

5

68

9

7

10

11121314

15 16

Figure 4.8 – A graph.

Let us instantiate the algorithm on the graph of Fig. 4.8. Given a matching M = {(2,3), (4,5),(6,8), (9,10), (11,12), (13,14), (15,16)}, an illustration of the detection of a M-blossom and thetermination of Edmonds’ algorithm with M a maximum matching are presented in Fig. 4.9.

Now, let us prove that the algorithm terminates with a maximum matching. Let M be the matchingin the last iteration of the algorithm and B1, ...,Bk the various blossoms discovered when building thecorresponding M-alternating forest. We will prove that the inner vertices of F ′ := F \B1 \B2 · · · \Bk(the alternating forest with shrinked blossoms), denoted by U form a Tutte set of the original graph.Observe that, by construction, super vertices are necessarily outer vertices. We denote by odd(G) the

69

b b

b b

b b

b b

b b

b

b

b

b

b

b

b

b b

b b

b

b b

b b

b b

b b

b b b

b

b

b b

b b

b

b

b

1

2

3

4

5

68

9

7

10

11121314

15 16

15 16

1314 12

8

9

7

10

6

11B1

1

24

5 3

7

B1

6 12

8 11

9

10

7

(a)A matching M and two exposed vertices 1 and 7. (b) Alternating forest detecting a blossom 1, 2, 3, 4, 5.

(c) A matching M and a contracted blossom B1. (d) An alternating forest detecting the end of Edmondsalgorithm, the matchcing M is maximum.

Figure 4.9 – Edmonds’ M-augmenting forest given a matching M.

inner vertices, by even(G) the outer vertices of G (all original vertices shrunk to a super vertex areconsidered outer vertices) and by free(G) the vertices of G that are neither inner vertices nor outervertices. We recap here some properties that we will exploit in the next section.

Property 4.1. At the end of Edmonds’ algorithm, we have the following properties :(i) odd(G) are vertices of F ′ (not super vertices).(ii) There is no edge of E between even(G) and f ree(G) and between vertices of even(G) that are

shrunk to a different super vertices of F ′.(iii) Vertices of odd(G) are matched by M to vertices in even(G).(iv) The edges of M induce a perfect matching in G[ f ree(G)].

Proof. We already observed (i), super vertices are only in even(G). Property (ii) is true otherwisewe are in one of the cases of the alternating forest construction listed above (case 1. or 2. or 3.)and the algorithm would not stop. Property (iii) is valid by the definition of inner vertices and outervertices. Property (iv) follows from the fact that vertices in f ree(G) are not exposed and the verticesin even(G)∪odd(G) are either exposed or matched to a vertex in even(G)∪odd(G).

Remember that U = odd(G), then it follows from (ii) that the connected component of G−Uare components of G[even(G)] or G[ f ree(G)]. Now G[even(G)] has exactly |V (F ′)|− |U | connectedcomponents as they are associated with the outer vertices of F ′. Now the connectivity within a supervertex is guaranteed by construction. The parity of super vertices is preserved by shrinking so thatthe connected component of G[even(G)] are of odd cardinality. Finally, the connected components ofG[ f ree(G)] are of even cardinality as M induces a perfect matching in G[ f ree(G)] by property (iv).Hence, counting two vertices per edge in M misses exactly o(G−U)−|U | vertices of V , the exposed

70

vertices of F ′ where o(G−U) is the number of odd components of G−U , and thus U is a Tutte setand the algorithm returns a maximum matching. This proves Theorem 4.3. Fig. 4.10 displays a Tutteset U = {6,9,12} corresponding to the graph G in Fig. 4.8.

b b

b b 1314

15 16

b b b9 6 12

b

b

b

b

b1

2

3

4

bb bb108 117

Figure 4.10 – Edmonds decomposition after finding a maximum matching M.

4.3.2 Network flows

In this section, we recall definitions and results related to network flows. Let G = (V,E) be adirected graph (in this case the edges are called arcs). Let s, t ∈V be, respectively, a source and a sinkof G: a source is a vertex with no incoming arcs and a sink is a vertex with no outgoing arcs. Let themapping c : E→ R+ be a capacity function on the arcs of G. The pair (G,c) is called a network, seeFig. 4.11 (a) for an example. A flow from s to t, called (s, t)−flow is a function f : E→R+ satisfyingthe following constraints:

— Capacity constraint: for each arc e ∈ E, the value of the flow going through e is at most thecapacity of the arc e i.e. 0≤ f (e)≤ c(e),∀e ∈ E.

— Conservation constraint: for each vertex v∈V \{s, t}, the sum of incoming flow to v is equalto the sum of outgoing flow from v i.e. ∀v ∈V \{s, t},∑e∈δ+(v) f (e)−∑e∈δ−(v) f (e) = 0.

The value of a (s, t)−flow noted | f | is equal to the sum of flows outgoing from the source vertex si.e. | f |= ∑e∈δ+(s) f (e). A maximum (s, t)−flow in G is a (s, t)−flow of maximum value, an exampleis presented in Fig. 4.11 (b). An integral (s, t)−flow is a flow that has an integer value that transits ineach arc of the network. A (s, t)−cut U is a set of vertices of V with s ∈U and t ∈V \U . The capacityof a cut U is the sum of the capacity on the arcs in δ+(U) i.e. the value ∑e∈δ+(U) c(e). A minimum(s, t)−cut in G is a (s, t)−cut of minimum capacity. An example of a (s, t)−cut is shown in Fig. 4.11(c) and (d). In [26], Ford and Fulkerson presented a direct relation between the maximum flow and theminimum cut of a network in the max-flow min-cut theorem, see Theorem 4.7 below. In addition, amaximum flow in a network can be found using Ford-Fulkerson method in O(|E| · | f ∗|) where | f ∗| isthe value of the maximum flow, see for instance [27], [13]. A polynomial time of the Ford-Fulkersonmethod was proposed by Edmonds-Karp [25]. The algorithm runs in time O(|V |.|E|2).

Theorem 4.7. (Ford Fulkerson theorem [26]). The value of a maximum (s, t)-flow is equal to thecapacity of the minimum (s, t)-cut.

71

bc

bc

bc

bc

bc bcs t2

a c

be

d

f

g

2

2

1

1

1

1

11

1

1

bc

bc

bc

bc

bc bcs t2

a c

be

d

f

g

2

2

1

1

1

1

11

1

1bc

bc

bc

bc

bc

bc

(s, t)−flow

b

bc

bc

b

b bcs t2

a c

be

d

f

g

2

0

1

1

1

1

11

1

1

b

bc

bc

(s, t)−cut U{s, a, b, c} are nodes in U

(a) A network with a source s and a sink t, (b) A (s, t)−flow of value 4

(d) (s− t) cut of value 4,

U = {s, a, b, c}, V \U = {e, d, f, g, t}.

the numbers are capacities

Figure 4.11 – A network, a max (s, t)-flow and a min (s, t)-cut.

4.4 k-matching cover

Now that we recalled the main results for matching and network flows, we can present our firstproof of Theorem 4.5.

Theorem 4.5. Let G = (V,E) be a graph with no perfect matching. Let k ≥ 2 be an integer. Thereexists a k-matching cover of G if and only if for all stable set S of G, |S| ≤ |N(S)| · k.

4.4.1 Proof of Theorem 4.5 (first version)

In this first proof we use Property 4.1. For convenience, at the end of Edmonds’ algorithm wedefine A(G) = odd(G), D(G) = even(G) and C(G) = f ree(G) . Let us denote by D1(G) the setof odd connected components of D(G) of size 1 and D3(G) := D(G) \D1(G). An illustration of thesets A(G),C(G),D(G),D1(G),D3(G) corresponding to the graph in Fig. 4.8 are presented in Fig. 4.12.

We will prove that the following statements are equivalent:— (a) ∃ k-matching cover of G.— (b) ∀S stable set of G, then |S| ≤ |N(S)| · k.— (c) ∃ a flow of size |D1| in the network (G′(V ′,E ′),c) where V ′ = {s, t}∪D1∪A , E ′ := {e =

(u,v) ∈ E : ∀u ∈ D1,v ∈ A}∪{(s,u) : ∀u ∈ D1}∪{(v, t) : ∀v ∈ A}, and c(e) = 1 if e = (s,u)for some u, c(e) = k if e = (v, t) for some v and c(e) = +∞ otherwise (an illustration is givenin Fig. 4.13).

(a) =⇒ (b) (Implication):Suppose that {M1, ...,Mk} is a k-matching cover of G. We denote by C the set

⋃ki=1 Mi. Let S be a

stable set of G. We let C′ := C∩ δ (S). First, by definition of C, for each vertex u ∈ S, there exists(u,v) ∈ C and because S is a stable set, v ∈ N(S). Hence, there are at least |S| edges in |C′| i.e.

72

b b

b b 1314

15 16

b b b9 6 12

b

b

b

b

b1

2

3

4

bb bb108 117

C(G)

A(G)

D(G)

admit a perfect matching in M

are matched by edges in M

from A(G) to D(G)

D3 admits a 2-matching cover

Does D1 admit a k−matching cover?

D1(G)

D3(G)

Figure 4.12 – Edmonds decomposition after finding a maximum matching M.

b

b

b

A(G)D1

bc bcs

t

≤ k≤ 1

∃(s− t) flow of value |D1| of the network G′?

b

b

b

A(G)D1

Bipartite graph withvertices in D1 ∪ A(G) and

edges (u, v) ∈ E s.t u ∈ D1 and v ∈ A(G)

(G′(V ′, E′), c)

b

b

b

b

b

b

b

b

+∞

Figure 4.13 – Example of a network (G′(V ′,E ′),c), as described in the above statement (c), corre-sponding to the Edmonds decomposition presented in Fig. 4.12.

|S| ≤ |C′|. Secondly, for each vertex v ∈ N(S), v has at most k incident edges in C and a fortiori in C′.Thus there are at most k · |N(S)| edges in C′ i.e. |C′| ≤ k · |N(S)|. The result follows.

(b) =⇒ (c) (Contrapositive):In the following, we will prove that if there is no flow of size |D1| in (G′ = (V ′,E ′),c) , then there

is a stable set S of G such that |S|> |N(S)| ·k. Let f ∗ be the value of a maximum (s, t)−flow of (G′,c).Let U be a minimum (s, t)−cut of G′, see Fig. 4.14. By the max-flow min-cut theorem, the capacityof the cut U is f ∗. Assume that f ∗ < |D1|. In particular, there is no arc of infinite capacity in δ+(U).It follows that :

1. f ∗ = |D1 \U |+ k · |A∩U |. Therefore f ∗ = |D1 \U |+ |A∩U | · k < |D1|, so |D1|− |D1∩U |+|A∩U | · k < |D1| and hence |A∩U | · k < |D1∩U |.

2. We have N+G′(D1 ∩U) ⊆ A∩U . Therefore |N+

G′(D1 ∩U)| · k ≤ |A∩U | · k < |D1 ∩U |. AndD1 ∩U is a stable set of G as the vertices in D1 are vertices from the original graph G (see

73

Property 4.1 (i)). Also N+G′(D1∩U) = N(D1∩U) for the same reason.

Thus S = D1∩U is a stable set of G such that |S|> |N(S)| · k.

bc bcs

t

+∞≤ 2

≤ 1

b

b

bbc

bc

bc

bc

A(G)D1(G)

≤ 1

≤ 1

≤ 1

≤ 2

≤ 2

+∞

+∞

+∞+∞

(s, t)−cut

U

bc+∞≤ 1

Figure 4.14 – U is an (s, t)−cut of value 4≤ |D1| of the network (G′ = (V ′,E ′),c). U ∩D1 is a stableset S such that |S|= 3 > 2×N(S) = 2.

(c) =⇒ (a) (Implication):Let f be a (s, t)-flow of value |D1| in (G′,c). We can always assume we have an integral flow for

integral capacities as shown in [13]. The value of the flow going through each arc of E[D1 : A] is 0or 1, because of the capacity of 1 on the arc leaving s in G′ and flow conservation at vertices in D1.Let C f = {(u,v) ∈ D1×A : f (u,v) = 1}. We will sometimes consider C f as a set of arcs in G′ andsometimes as a set of edges in G. Because f is of value |D1| then each vertex in D1 is incident toexactly one arc in C f . Similarly each vertex in A is incident to at most k arcs in C f . Therefore theconnected components of the (bipartite) subgraph G of G with vertex set D1∪A and edge set C f arestars centered in A with degree at most k. Hence, C f can be decomposed greedily into k-matchingsM1, . . . ,Mk of G : number the edges of each star from 1 to k and assign the edge numbered i to thematching Mi. Observe that {M1, . . . ,Mk} is a k-matching cover of G that covers each vertex of D1exactly once. Note also that some Mi might be empty.

Let A′ ⊆ A be the set of vertices that are not covered by edges in C f . Let M be the maximummatching of G returned by Edmonds’ algorithm. M′ = M \ δ (A \A′) is a matching of G that coversall vertices in A′ (by Property 4.1 (iii)), all vertices in C (by Property 4.1 (iv)) and all non exposedvertex in D3. In other words, the edges of M′ are in E[C], E[D3], E[A′ : D3] or E[A′ : D1] by Property4.1. Now let us complete the k matchings M1, . . . ,Mk to cover all the vertices of G using the matchingM without generating any conflicts. For each e ∈M′, we assign e to the first Mi such that Mi∪{e} isstill a matching of G. Observe that conflicts can appear only between edges of M \M′ and edges inthe original Mi,∀i ∈ {1, . . . ,k} as M \M′ and the laters are matchings. We thus have the following :

— Edges of M in E[C] are assigned to M1 : no conflicts is ever created.— Edges of M in E[D3] are assigned to M1 : no conflicts is ever created.— Edges of M in E[A′ : D3] are assigned to M1 : no conflicts is ever created.— Edges of M in E[A′ : D1] are assigned to M1 or M2 (remember that k ≥ 2) : indeed an edge

e ∈M∩E[A′ : D1] has at most one extremity u in D1 and each vertex of D1 is covered exactlyonce by ∪k

i=1Mi, so either M1 misses u or M2 misses u.

The new Mi, 1 ≤ i ≤ k, cover all vertices of G but the exposed vertices in D3. By property ofthe M-alternating forest F ′, there is only one exposed vertex per component of G[D]. Now, for each

74

exposed vertex u ∈ D3, there is an edge in N(u)∩D3 (otherwise u would be in D1). We add this edgeto M2. The corresponding edges do not share a vertex as each exposed vertex of D3 is in a differentconnected component of G[D3] and the edges of M in E[D3] were assigned to M1. Thus M2 is still amatching. Now {Mi,1≤ i≤ k} is a k-matching cover of G and we are done.

4.4.2 Proof of Theorem 4.5 (second version)

We give here an alternating (and simpler) proof of the same result that builds essentially uponaugmenting path technique in bipartite graphs and does not use the matching of Edmonds algorithm.

Proof. We already proved the « if » part of the theorem: see the proof (a)→ (b) in 4.4.1. Now let usprove the « only if » part of the theorem. Using Claim 1 presented below, it is enough to prove that ifthere does not exist F ⊆ E such that 1≤ dF(v)≤ k, where dF(v) is the number of edges in F incidentto v, then it exists a stable set S such that |S|> |N(S)| · k.

Claim 1: ∃k−matching cover of G ⇐⇒ ∃F ⊆ E such that 1≤ dF(v)≤ k.

⇒ let M1, . . . ,Mk be the set of matchings of G in the k−matching cover for all v∈V , v is incidentto at most k edges in ∪i=1...kMi. Therefore, let F be the union of the k matchings i.e. F = ∪i=1...kMi,we have 1≤ dF(v)≤ k. ⇐ assume now that there exists F ⊆ E such that 1≤ dF(v)≤ k. While thereis an edge (u,v) ∈ F such that both u and v are covered by at least two edges of F , delete (u,v) from Fand let us call this new set of edges F ′. Then G(V,F ′) is a forest of stars with at most k edges. Let π

be a mapping of the edges of F ′ to {1, . . . ,k} so that no two edges incident to the same vertex receivethe same label. Then, M1 = π−1(1), . . . ,Mk = π−1(k) is a k−matching cover.

Let B be the set of vertices incident to at most 1 edge of F , A := V \B the rest of the vertices ofG, moreover, let B′ ⊆ B be the set of vertices of V not covered by F . Suppose that the set B′ 6= /0. Weshow then that it exists a stable set S such that |S|> |N(S)| · k .

Let A′ be the set of vertices of A reachable from B′ by an alternating path, where the vertices of thepath are alternating between B and A and the edges of the path are alternating between E \F and F . LetS be the set of vertices of B reachable from B′ by an alternating path where B′ ⊆ S⊆ B. Note that thesepaths are alternating paths in the bipartite graph induced by edges with one endpoint in A, the otherin B. It is well known how to determine A′ and S for such bipartite graphs with classes A and B algo-rithmically: it is actually just accessibility in a directed graph that can be found by Breath First Search.

Claim 2: The vertices in A′ are incident to exactly k edges of F.

Indeed, if a vertex in A′ incident to less than k edges of F is reachable from B′ by an alternatingpath, then we can increase the number of vertices covered by interchanging the edges in F and not inF along this alternating path.

Claim 3: S is a stable set and N(S) = A′.

Indeed, there is no edge from S to any vertex of B: this assertion is true for B′, because addingsuch an edge would increase the number of covered vertices; there is also no edge from S \B′ to anyB \B′ vertex, otherwise if we have a (s, t) edge such that s ∈ S \B′ and t ∈ B \B′, then we get, byinterchanging along a (b,s) alternating path (b ∈ B′) and after adding the edge (s, t) to F , a solutionwith one more vertex covered (vertex b). So the assertion is also true for them, and the first part ofthe claim follows. There is also no edge from S to any vertex a ∈ A\A′ since the existence of such anedge would prove a ∈ A′. The claim follows.

75

By Claim 2 and Claim 3, |S\B′|= k|A′|, and since B′ 6= /0, |S|> k|N(S)|.

4.5 Conclusion

In this chapter we studied a special case of the DSA-aware MP problem that can be solved in poly-nomial time. Our results present two proofs of the existence of a k−matching cover. Both proofs canbe straightforwardly turned to algorithms, the first proof yields an algorithm with the same complexityas Edmonds’ algorithm, while the second proof yields an algorithm that has the same complexity asbipartite matching. This study is more theoretical than practical because it is unlikely for the instancesfrom the application to meet the conditions of the reduction to the k−matching cover problem. Indeed,it is unlikely for the auxiliary graph G′ that list all the feasible groups of an industrial instance to be aline graph.

76

Conclusion and perspectives

In this thesis, we developped exact methods to solve the DSA-aware Multiple Patterning problem.We proposed two graph coloring models: a general k−path coloring problem and an induced k−pathcoloring problem. These two problems are known to be NP-hard and we proposed three differentapproaches to solve them. First, we developed and compared several integer linear programming for-mulations. Second, we investigated the characteristics of the graphs arising from industrial instances:they have « typically » a small tree-width and we exploited this property to develop a dynamic pro-gramming approach. Finally, we studied a special case when the graphs are line: in this case theDSA-aware MP problem reduces to the k−matching cover problem. Further research could be doneon the three different approaches that we studied as detailed below.

In Chapter 2, experiments were done on the several ILP formulations to solve the general andinduced k−path coloring, using CPLEX solver. Because the current DSA technology allows only todeal with small values of k (1 or 2), we restricted our experiments to this setting (even though we de-velopped models capable of handling larger k). It is not too surprising that the naïve ILP formulationperforms best for the small values of k. However, we are expecting that this latter formulation willloose interest when the value of k increases, because the number of paths of length at most k growsexponentially with k. It would be important to run further tests with larger values of k and make newcomparisons between the proposed formulations to identify the turning point. We also observed thatthe general k−path coloring problem was harder to solve than the induced one. As any solution to theinduced problem is a solution to the general one, we could use an optimal solution to the former as aninput to the latter and see if the runtime will be improved.

To evaluate the interest of a column generation approach, we have chosen to use BaPCod frame-work [38]. We did preliminary experiments for the induced 1−path coloring problem on the industrialinstances. The results showed that BaPCod had a lot of difficulties to solve the ILP formulations com-pared to CPLEX. One could investigate further such an approach by varying the parameters used byBaPCod and see if there is a positive impact on the results (according to Dr Ruslan Sadykov, the re-sults could change with different parameters, so we should be careful not to draw quick conclusions).As BaPCod is a generic tool, we could also try to use a column generation algorithm tailored to theproblem.

One of the main objectives of Mentor Graphics in this project is to assess the quality of their heuris-tics. We also had some interest in developping alternative heuristics and also assess their quality withour tools . Tests were done on the induced k−path coloring when k = 1 and k = 2 with LocalSolver.LocalSolver found optimal solutions in few seconds for all industrial instances, however no certificateof optimality was returned after one hour. In practice, our first experiments show that LocalSolvermight be a good heuristic that might improve upon the current heuristics used in the industry for theDSA-aware MP problem. Besides, the solution retrieved by LocalSolver might be useful to speed-upthe exact resolution with CPLEX. This is another direction that would be interesting to explore.

77

When we received our first sample of industrial instances, we realized that they all had a smalltreewidth. In Chapter 3, we decided to exploit this property and we developped a dynamic program-ming approach to solve the induced k−path coloring when k = 1 and k = 2. Our experimental resultsshowed similar performances as with the naïve ILP formulation implemented using CPLEX. Furtherinvestigations could be done using the dynamic programming approach. The algorithms that we pre-sented for the induced k−path coloring, where k = 1 and k = 2, could be extended to larger values of k.Similar algorithms could also be developped for the general k−path coloring problem. Depending onthe results found when the value of k increases, one could also try to bypass D-FLAT and implementthe dynamic programming directly to further improve the runtime (if needed).

In Chapter 4, we studied the special case where the graphs are line and we reduced the DSA-awareMP problem to the k−matching cover problem and we could solve it in polynomial time by extendingtechniques from bipartite matching. We might try to use the augmenting path idea as a heuristic tosolve the DSA-aware MP problem in the general case. Moreover, one could also do further investiga-tions on the characteristics of the graphs arising from industrial instances. We might then find relevantcases where the k−path coloring can be solved in polynomial time.

Finally, we could also think about improving our models models to integrate other meaningfulmanufacturing constraints to the problem. As discussed in the introduction chapter, our formulationscan be used to solve DSA-aware MP in the case of EUV lithography as well. In this case though,we need to add the angles constraints to the formulations. We could also add weights on the edgesin order to favor small groups, in the optimal solutions, that are preferred by the DSA process (suchpreferences, indeed exist in practice). Again this can be easily added to our formulations.

The industry showed an intense interest in DSA technology in the past years. In particular, Mentorgraphics want to be part of the first companies delivering these promising solutions in the case whereDSA technology started to be used for large scale manufacturing. However, with the arrival of EUV,DSA has lost interest lately. However given the pressure in being able to manufacture smaller andsmaller components, it is likely that DSA will find interest again. We hope that the tools we developpedin this thesis will help the DSA community to grow and Mentor Graphics to position itself as animportant player.

78

Contributions

This PhD thesis was developed at Mentor Graphics Corporation in collaboration with G-SCOPlaboratory, Grenoble, France. This work was part of the European project PLACYD, that was sup-ported by the ENIAC Joint Undertaking under grant agreement n◦226927.

79

Bibliography

[1] M. S. A. Graf and G. Weissenfels. On coloring unit disk graphs. Algorithmica, 18:277–293,1998.

[2] A.A.Zykov. On some properties of linear complexes. Mat. Sbornik, 24:313–319, 1949. inRassian.

[3] S. Ahmed. Applications of graph coloring in modern computer science. IJCIT, 03(02), 2012.

[4] K. Appel and W. Haken. Every planar map is four colourable. i. discharging. Illinois J.Math,21:429–490, 1977.

[5] K. Appel and W. Haken. Every planar map is four colourable. i. discharging. Illinois J.Math,21:429–490, 1977.

[6] K. Appel and W. Haken. Every planar map is four colourable. ii. reducibility. ContemporaryMathematics 98. American Mathematical Society, Providence, RI, 1989.

[7] M. Aslan and N. A. Baykan. A performance comparison of graph coloring algorithms. Interna-tional Conference on Advanced Technology and Sciences (ICAT 16), pages 266–273, 2016.

[8] Y. Badr, A.Torres, , and P.Gupta. Mask assignment and synthesis of dsa-mp hybrid lithographyfor sub-7nm contacts/vias. In Proceedings of the 52Nd Annual Design Automation Conference,DAC ’15, pages 70:1–70:6, New York, NY, USA, 2015. ACM.

[9] X.-Y. Bao, H. Yi, C. Bencher, L.-W. Chang, P.-T. C. H. Dai, Y. Chen, and H.-S. Wong. Sram,nand, dram contact hole patterning using block copolymer directed self-assembly guided bysmall topographical templates.

[10] C. Berge. Sur le couplage maximum d’un graphe. Comptes rendus hebdomadaires des séancesde l’Académie des sciences, 247:258–259.

[11] C. Berge. Two theorems in graph theory, volume 43. Proceedings of the national academy ofscience of the United States of America, 1957.

[12] C. Berge. färbung von graphen, deren sämtliche bzw. deren ungerade kreise starr sind. Wiss.Z.Martin-Luther-Univ.Halle-Wittenberg Math.-Natur, 10, 1961.

[13] B.Korte and J.Vygen. Combinatorial Optimization: Theory and Algorithms. Springer PublishingCompany, Incorporated, 5th edition, 2011.

[14] H. L. Bodlaender and A. M. Koster. Treewidth computations i. upper bounds. Information andComputation, 208(3):259 – 275, 2010.

[15] H. L. Bodlaender and A. M. C. A. Koster. Combinatorial optimization on graphs of boundedtreewidth. Oxford University Press, 2007.

[16] G. Brewka, T. Eiter, and M. Truszczynski. Answer set programming at a glance. ACM, 54, 2011.

[17] R. Brooks. On coloring the nodes of a network. Proc. Cambridge Philo, 37:194–197, 1941.

[18] L. Chang, X. Bao, C. Bencher, and H. Wong. Experimental demonstration of aperiodic patternsof directed self-assembly by block copolymer lithography for random logic circuit layout. IEEE,pages 33.2.1–33.2.4, 2010.

80

[19] N. Chmait and ve K.Challita. Using simulated annealing and ant-colony optimization algorithmsto solve the scheduling problem. Computer Science and Information Technolog, 3:208–224,2013.

[20] U. Congress and O. of Technology Assessment. Miniaturization technologies. OTA-TCT-514,Washington, DC: U.S. Government Printing Office, November 1991.

[21] B. Courcelle. The monadic second order logic of graphs. i. recognizable sets of finite graphs.Information and Computation, 85:12–75, 1990.

[22] K. Dowsland and J. Thompson. Ant colony optimization for the examination scheduling prob-lem. J. Oper. Res. Soc, 56:426–438, 2005.

[23] D.-Z. Du and P. P. (Eds.). Handbook of Combinatorial Optimization, volume Vol. 2. KLUWERACADEMIC PUBLISHERS, 1998.

[24] J. Edmonds. Paths, trees, and flowers, volume 17. Canad. J. Math., 1965.

[25] J. Edmonds and R. M. Karp. Theoretical improvements in algorithmic efficiency for net-work flow problems. Journal of the ACM, Association for Computing Machinery (ACM),19(2):248–264, 1972.

[26] L. Ford and D. Fulkerson. Maximal flow through a network. Canadian journal of Mathematics,8(3):399–404, 1956.

[27] L. R. Ford and D. R. Fulkerson. Flows in Networks. Princeton University Press, 6th edition,1974.

[28] M. R. Garey and D. S. Johnson. Some simplified np-complete graph problems. theoriticalcomputer science, pages 237–267, 1976.

[29] R. Griffiths, A. Williams, A. V. C. Oakland, J. Roberts, and T. Thomson. Directed self-assemblyof block copolymers for use in bit patterned media fabrication. IOPscience, 2013.

[30] M. Grötschel, L. Lovász, and A. Schrijver. Geometric algorithms and combinatorial optimiza-tion. Springer-Verlag, pages 273–303, 1988.

[31] J. Hahm. Functional polymers in protein detection platforms: Optical, electrochemical, electri-cal, mass-sensitive, and magnetic biosensors. Sensors, pages 3327–3355, 2011.

[32] W. Hale. Frequency assignment: Theory and applications. IEEE Proceedings, 68(12):1497–1514, dec 1980.

[33] P. Hall. On representatives of subsets. J. London Math. Soc, 10:26–30, 1935.

[34] F. Havet, R. J. Kang, and J.-S. Sereni. On improper colouring of unit disk graphs. Networks,pages 150–164, 2009.

[35] X.-Y. B. He Yi, J. Zhang, C. Bencher, L.-W. Chang, X. Chen, R. Tiberio, J. Conway, H. Dai,Y. Chen, S. Mitra, and H.-S. P. Wong. Flexible control of block copolymer directed self-assemblyusing small, topographical templates: Potential lithography solution for integrated circuit contacthole patterning. Advanced materials, 24, 2012.

[36] I. Holyer. The np-completeness of edge-colouring. SIAM J. COMPUT, 10:718–720, 1981.

[37] I.Karageorgos, J.Ryckaerta, R.Gronheida, M.C.Tungc, H.-S.P.Wongc, E.Karageorgosd,K.Croesa, J.Bekaerta, G.Vandenberghea, M.Stucchia, and W.Dehaeneb. Design method andalgorithms for directed self-assembly (dsa) aware via layout decomposition in sub-7 nm circuits.Journal of Micro/Nanolithography, Microfabrication, and Microsystems, 15, 2016.

[38] Inria. Bapcod - a generic branch-and-price code, 2005.

[39] Intel. Sand to silicon - the making of a chip (full screen) | intel.

81

[40] N. D. Jarnagin. High χ block copolymers for sub 20 nm pitch patterning: synthesis, solventannealing, directed self assembly and selective block removal. PhD thesis, Georgia Institute ofTechnology, 2013.

[41] Y. Jinjiang. Computational complexity of (2,2)path chromatic number problem. Appl. Math,10:83 – 88, 1995.

[42] Y. Jinjiang and L. Yixun. Some results about path chromatic numbers of graphs. J. ZhengzhouUniv, 21:4:1 – 8, 1992.

[43] J.Mycielski. Sur le coloriage des graphs. Onformation Processing Letters, 108:412–417, 2008.

[44] D. Johnson. Worst case behavior of graph coloring algorithms. Proc. 5th Southeastern Conf. onCombinatorics, Graph Theory and Computing, pages 513–527, 1974.

[45] R. Karp. Reducibility among combinatorial problems. Complexity of computer computations(Plenum Press, New York), pages 85–104, 1972.

[46] D. Konig. Gráfok és mátrixok. Matematikai és Fizikai Lapok, 38:116Ð119, 1931.

[47] T. Kloks. Treewidth: Computations and approximations. Springer-Verlag, 1994.

[48] J. Kuang, J. Ye, and E. F. Young. Simultaneous template optimization and mask assignment fordsa with multiple patterning. Design Automation Conference (ASP-DAC), 21st Asia and SouthPacific, 2016.

[49] J. Kuang and E. F. Y. Young. An efficient layout decomposition approach for triple patterninglithography. ACM 978-1-4503-2071-9/13/05, 2013.

[50] H. W. Kuhn. The hungarian method for the assignment problem. Naval Research LogisticsQuarterly, 2:83–97, 1955.

[51] W. A. Lopes and H. M. Jaeger. Double-patterning-compliant split and design. Nature, 414:735–738, 2011.

[52] L. Lovász and M. D. Plummer. Matching theory. North-Holland mathematics studies. North-Holland, Amsterdam, New York, 1986.

[53] M.Abseher, B.Bliem, G.Charwat, F.Dusberger, M.Hecher, and S.Woltran. D-flat: progress re-port. DBAI Technical report, Vienna University of technology, 2014.

[54] M.Chudnovsky, N. Robertson, P. Seymour, and R. Thomas. The strong perfect graph theorem.manuscript, 2002.

[55] M.Chudnovsky, N. Robertson, P. Seymour, and R. Thomas. The strong perfect graph theorem.Annals of mathematics, 164:51 – 229, 2006.

[56] M. Mirsaeedi. EDA Solutions for Double Patterning Lithography. PhD thesis, University ofWaterloo, Ontario, Canada, 2012.

[57] G. Moore. Cramming more components onto integrated circuits. Electronics, 38, 1965.

[58] W. Moreau. Semiconductor Lithography: Principles, Practices, and Materials. Microdevices(New York, N.Y). Plenum Press, 1988.

[59] N. Musliu. Recent Advances in Evolutionary Computation for Combinatorial Optimization.Studies in Computational Intelligence, volume 153. Springer, Berlin, Heidelberg, 2008.

[60] N.Robertson and P. Seymour. Graph minors. ii. algorithmic aspects of tree-width. J. Algorithms,7:309–322, 1986.

[61] M. Peeters. On coloring j-unit sphere graphs. Research memorandum / Tilburg University,Departement of Economics, 1991.

[62] N. Robertson, D. P. Sanders, P. D. Seymour, and R. Thoma. A new proof of the four colourtheorem. Electron. Res. Announc. Amer. Math. Soc, 2:17–25, 1996.

82

[63] S.Arnborg, D. G. Corneil, and A. Proskurowski. Complexity of finding embedding in a k-tree.SIAM J. Alg. Disc. Meth., 8:277–284, 1987.

[64] A. Schrijver. Matroids in linking systems. J. Combin., 1979.

[65] W. T. Tutte. The factorization of linear graphs. J. London Math. Soc., 22:107–111, 1947.

[66] V.G.Vizing. Critical graphs with given chromatic class. Diskret, Analiz, 5, 1965.

[67] X. Wang, X. Song, and J. Yuan. On matching cover of graphs. Mathematical Programming,147(1):499–518, 2014.

[68] X. Wang, X. Song, and J. Yuan. A note on matching cover algorithm.https://arxiv.org/pdf/1612.00943.pdf, December, 2016.

[69] Y. Wei and D. Back. 193nm immersion lithography: Status and challenges. SPIE, 2007.

[70] Yang, Yunfeng, Luk, Wai-Shing, Zhou, Hai, Pan, D. Z., Zhou, Dian, Yan, Changhao, Zeng, andXuan. An effective layout decomposition method for dsa with multiple patterning in contact-holegeneration. ACM Trans. Des. Autom. Electron. Syst., 23(1):11:1–11:27, 2017.

[71] Y.Badr, J.A.Torres, and P.Gupta. Incorporating dsa in multipatterning semiconductor manufac-turing technologies. SPIE Advanced technology, 2015.

[72] B. Yu, K. Yuan, B. Zhang, D. Ding, and D. Z. Pan. Layout decomposition for triple patterninglithography. In Pro. ICCAS, 2011.

83

Appendix A

Definitions

Definition A.1. (Graph). An undirected graph G is a pair (V,E) where V is a finite set of nodes orvertices and E is a set of pairs of nodes called edges.

See (a) in Fig. A.1 for some graphs examples. Nodes that define an edge are called extremities ofthe edge. Two nodes linked by an edge are called adjacent. Two edges sharing a same extremity arecalled incident. In our study we consider graphs without any loops (i.e. no edge of the type (u,u)).

Definition A.2. (Subgraph). The subgraph G′ = (V ′,E ′) of a graph G = (V,E) is a graph whereV ′ ⊆V and E ′ ⊆ E and ∀(u,v) ∈ E ′, u ∈V ′ and v ∈V ′. See (b) in Fig. A.1.

Definition A.3. (Induced subgraph). Let G = (V,E) be a graph. Let V ′ ⊆V and let E ′ ⊆ E. A graphG′ = (V ′,E ′) is an induced subgraph of G if and only if it is a subgraph of G and ∀u ∈ V ′, ∀v ∈ V ′,(u,v) ∈ E→ (u,v) ∈ E ′. The subgraph induced by V ′ of G is denoted by G[V ′], see (c) in Fig. A.1.

Definition A.4. (Stable set). Let G = (V,E) be a graph. Let S⊆V be a set of nodes of G. S is a stableset if and only if the induced subgraph G[S] has no edge, i.e. no two nodes in S are adjacent in G. (i.e∀u,v ∈ S,(u,v) /∈ E), see (d) and (e) in Fig. A.2.

1

23

6 7

8

4

5

1

23

6 7

4

1

23

6 7

(a) A graph G(V,E), (b) A subgraph G′(V ′, E′) of G (c) An induced subgraph G[V ′] of G

4

V = {1, 2, 3, 4, 5, 6, 7, 8},E = {(1, 2), (1, 3), (2, 3), (2, 7),

(2, 8), (7, 8), (3, 6), (6, 7),

(3, 4), (4, 6), (3, 5), (4, 5), (5, 6)}

V ′ = {1, 2, 3, 4, 6, 7},E′ = {(1, 2), (1, 3), (3, 4), (6, 7)}

V ′ = {1, 2, 3, 4, 6, 7},

Figure A.1 – Example of a graph , a subgraph of a graph and an induced subgraph of a graph.

84

1

23

6 7

8

4

5

1

23

6 7

4

(d) A stable set S = {1, 5, 7} (f) A path

1

23

6 7

8

4

5

(e) S′ = {1, 3, 4, 8} is not a stable set

Figure A.2 – Example of a stable set of nodes in a graph and a path.

Definition A.5. (Path graph). Let G = (V,E) be a graph. G is a path graph if the set of nodes V canbe written in a sequence v1, . . . ,vn and E = {(vi,vi+1)∀i ∈ {1, . . . ,n−1}}. A path contains two nodeswith degree 1 and all the other nodes have degree 2. The length of the path G is the number of edgesin E. See (f) in Fig. A.2.

Definition A.6. (Degree of nodes). Let G = (V,E) be a graph. Let v∈V , the degree of v is the numberof edges incident to v denoted deg(v).

Definition A.7. (Maximum degree). Let G = (V,E) be a graph. The maximum degree of a graphG is the maximum degree over all degrees of nodes in V . It is denoted by ∆(G). i.e ∆(G) =max{deg(v),∀v ∈V}.

Definition A.8. (Connected component). A connected component of a graph G = (V,E) is a subgraphof G such that ∀u,v ∈V there exists a path in G from u to v.

Definition A.9. (Line graph). A line graph L(G) of a graph G is defined as follows. The vertices ofL(G) correspond to the edges of G, and two vertices of L(G) are adjacent if and only if the corre-sponding edges are incident in G .

Definition A.10. (Planar graph). A planar graph G is a graph that can be drawn in the plane in sucha way that the edges of the graph do not intersect: they can only meet at their endpoints.

Definition A.11. (Complete graph). A complete graph G = (V,E) is a graph where every node u ∈Vis adjacent to every in V\{u}.

Definition A.12. (Bipartite graph). A bipartite graph G = (V,E) is a graph where the set of nodescan be partitioned into two disjoint sets of nodes, i.e. V = V1 ∪V2 and V1 ∩V2 = /0, such that everyedge (u,v) ∈ E has one extremity in V1 and the other extremity in V2.

In the case of undirected graphs, we consider the following notations: we denote a bipartite graphwith G = (X ,Y,E) such that the vertex set of G is X ∪Y and the edge set of G is E ⊆ X×Y .

For each set S⊆V (G), we denote:

85

— N(S): the set of neighbors of vertices of S i.e.{v ∈V (G)\S : ∃u ∈ S with (u,v) ∈ E}.— δ (S): the set of edges with one extremity in S and the other in N(S).For any S1,S2 ⊆V (G), we denote:— G[S1]: the subgraph of G induced by vertices in S1.— G−S1: the subgraph of G induced by the vertices in V (G)\S1, i.e. G[V \S1].— E[S1 : S2]: the subset of edges having one extremity in S1 and one extremity in S2.— E[S1]: the subset of edges having the two extremities in S1 i.e. E[S1 : S1].

In the case of directed graphs, we consider the following notations. For each vertex v ∈ V , wedenote:

— δ+(v): the set of outgoing arcs of v i.e. {(v,w) ∈ E}.— δ−(v): the set of incoming arcs to v i.e. {(u,v) ∈ E}.

For the set of vertices U ⊆V , we define:— δ+(U): the set of outgoing arcs of U i.e. {(u,v) ∈ E : u ∈U,v 6∈U}.— N+(U): the set of neighbors of vertices of U outgoing from U i.e. {v∈V \U :∃u∈U s.t. (u,v)∈

E}.— δ−(U): the set of incoming arcs to U i.e. {(v,u) ∈ E : u ∈U,v 6∈U}.— N−(U): the set of neighbors of vertices adjacent to U going toward U i.e. {v ∈ V \U : ∃u ∈

U s.t. (v,u) ∈ E}.

Sometimes to avoid confusion we index all previous notations by G, for instance we denote N(S)by NG(S), δ (S) by δG(S) and so on. In addition, let us recall some basic definitions and properties ofmatching in a graph.

86

Appendix B

Algorithm for grid graphs

87

Data: probability p ∈ [0,1], matrix_size n ∈ NResult: Graph G(nodes, edges)edges = [ ], nodes = [ ], m = [ ] ;# create a matrix of size n∗n ;for i in range(n) do

m.append([0]∗n);endfor i in range(n) do

for j in range(n) do# choose a random number within [0,1] ;r = random.random() ;if r < p then

m[i][ j] = 1 ;nodes.append((n∗ i)+ j) ;

elsem[i][ j] = 0 ;

endif i≥ 0 and j ≥ 1 then

if m[i][ j−1] == m[i][ j] == 1 thenedges.append([(n∗ i)+( j−1) , (n∗ i)+ j, 1]) ;

endendif i≥ 1 and j ≥ 0 then

if m[i−1][ j] == m[i][ j] == 1 thenedges.append([(n∗ (i−1))+( j) , (n∗ i)+ j, 1]) ;

endendif i≥ 1 and j ≥ 1 then

if m[i−1][ j−1] == m[i][ j] == 1 thenedges.append([(n∗ (i−1))+( j−1) , (n∗ i)+ j, 1.4]) ;

endendif i≥ 0 and j ≥ 2 then

if m[i][ j−2] == m[i][ j] == 1 thenedges.append([(n∗ (i))+( j−2) , (n∗ i)+ j, 2]) ;

endendif i≥ 2 and j ≥ 0 then

if m[i−2][ j] == m[i][ j] == 1 thenedges.append([(n∗ (i−2))+( j) , (n∗ i)+ j, 2]) ;

endendif i≥ 1 and j < (n−1) then

if m[i−1][ j+1] == m[i][ j] == 1 thenedges.append([(n∗ (i−1))+( j+1) , (n∗ i)+ j, 1.4]) ;

endend

endendOutput: nodes, edges ;

Algorithm 1: Algorithm for generating grid graph88

Appendix C

D-FLAT implementation

In this appendix we give the implementation of dynamic programming for the k−path coloringwhen k = 1 and k = 2 using D-FLAT framework.

Given a graph G(V,E) and F ⊆ E, we fix a set of colors and test if nodes of the graph can becolored with that set of colors. In the following sections we present:

— The input file format noted: input.lp, where vertex(i) defines every vertex i∈V and dsa_edge(u,v)define every edge (u,v) ∈ F and non_dsa_edges(u,v) define every edge in E\F .

— The code file of the 1-path coloring using a set of 3 colors noted: code_1.lp.— The code file of the 2-path coloring using a set of 2 colors noted: code_2.lp.

C.1 Input format

vertex(1). vertex(2). vertex(3). vertex(4). vertex(5).dsa_edge(2, 5). dsa_edge(3, 4). non_dsa_edges(1, 2). non_dsa_edges(1, 5). non_dsa_edges(2, 3).non_dsa_edges(4, 5).

C.2 D-FLAT code for 1-path coloring problem

color(red;green;blue).% non directed graph

dsa_edge(X,Y) :- dsa_edge(Y,X).non_dsa_edge(X,Y) :- non_dsa_edge(Y,X).

% guess rows to be extended1 extend(R) : childRow(R,N) 1 :- childNode(N).

% information contained in a child row has to be retained in case that the corresponding vertex isstill present in the current nodeitem(map(X,C)) :- extend(R), childItem(R,map(X,C)), current(X).

% check whether rows coincide on vertices receiving same colors:- extend(R), extend(S), R!=S, childItem(R, map(X,C1)), childItem(S, map(X,C2)), C1!=C2.

% for each introduced vertex we guess a color1 item(map(X,C)) : color(C) 1 :- introduced(X).

89

% we remove every solution that assigns same color to nodes X,Y where non_dsa_edge(X,Y):- non_dsa_edge(X,Y), item(map(X,C)), item(map(Y,C)).

% additionally use function symbol f/1% for any selected vertex X, f(X) is derived if X has a dsa neighbour with the same coloritem(f(X)) :- item(map(X,C)), item(map(Y,C)), dsa_edge(X,Y), X!=Y.item(f(X)) :- extend(R), childItem(R, f(X)), current(X).

% check an introduced vertex X do not receive same color as a current vertex Y such that Y al-ready has a dsa neighbour with the same color in child node (do not create a chain with 3 nodes whereX is an extremity).:- item(map(X,C)), extend(R), childItem(R, map(Y,C)), childItem(R, f(Y)), dsa_edge(X,Y), intro-duced(X), current(Y).

% check an introduced vertex X do not receive same color as two current vertices Y, Z such thatY and Z have same color but not necessary a dsa edge (do not create a chain with 3 nodes where X isin the middle).:- item(map(X,C)), item(map(Y,C)), item(map(Z,C)), dsa_edge(X,Y), dsa_edge(X,Z), Y!=Z, Y!=X,X!=Z, current(Y), current(Z), introduced(X).

% check that two joined rows R and S do not contain a node X such that X has a dsa neighbourwith same color that appears in child R and child S.:- extend(R), extend(S), R!=S, childItem(R, f(X)), childItem(S, f(X)), childItem(R, map(X,C)), childItem(S,map(X,C)), N1 = #countY: childItem(R,map(Y,C)), dsa_edge(X,Y), N1==0, N2 = #countY: childItem(S,map(Y,C)),dsa_edge(X,Y), N2==0.

# show item/1.

# show extend/1.

C.3 D-FLAT code for 2-path coloring problem

color(red;green).

%non directed graphdsa_edge(X,Y) :- dsa_edge(Y,X).non_dsa_edge(X,Y) :- non_dsa_edge(Y,X).

%guess rows to be extended1 extend(R) : childRow(R,N) 1 :- childNode(N).

%information contained in a child row has to be retained in case that the corresponding vertex isstill present in the current nodeitem(map(X,C)) :- extend(R), childItem(R,map(X,C)), current(X).

%check whether rows coincide on vertices receiving same colors:- extend(R), extend(S), R!=S, childItem(R, map(X,C1)), childItem(S, map(X,C2)), C1!=C2.

90

%for each introduced vertex we guess a color1 item(map(X,C)) : color(C) 1 :- introduced(X).

%we remove every solution that assigns same color to nodes X,Y where non_dsa_edge(X,Y):- non_dsa_edge(X,Y), item(map(X,C)), item(map(Y,C)).

%additionaly use function symbol f/1%for any selected vertex X, f1(X) is derived if X has a dsa neighbour with the same coloritem(to(X,Y)) :- item(map(X,C)), item(map(Y,C)), dsa_edge(X,Y), X!=Y.

item(path(X,Z)) :- item(map(X,C)), extend(R), childItem(R, map(Y,C)), childItem(R, to(Y,Z)),dsa_edge(X,Y), not dsa_edge(X,Z), introduced(X), current(Y), X!=Y, X!=Z, Y!=Z.

item(path(Y,Z)) :- item(map(X,C)), extend(R), childItem(R, map(Y,C)), childItem(R, map(Z,C)),dsa_edge(X,Y), dsa_edge(X,Z),not dsa_edge(Y,Z), Y!=Z, Y!=X, X!=Z, current(Y), current(Z), intro-duced(X).

item(path(X,Z)):- item(path(Z,X)).

item(to(X,Y)) :- extend(R), childItem(R, to(X,Y)), current(X).item(path(X,Z)) :- extend(R), childItem(R, path(X,Z)), current(X).

%forbidden cases for introduced node:- item(map(X,C)), extend(R), childItem(R, map(Y,C)), childItem(R, path(Y,Z)), dsa_edge(X,Y), in-troduced(X), current(Y).

:- item(map(X,C)), extend(R), childItem(R, map(Y,C)), N = #countZ: childItem(R,to(Y,Z)), N==2,dsa_edge(X,Y), introduced(X), current(Y), X!=Y.

% forbidden cases for join nodes:- extend(R), extend(S), R!=S, childItem(R, path(X,Y)), childItem(S, path(X,Y)), N1 = #countZ:childItem(R,to(X,Z)), N1<=1.

%other forbidden configurations:- item(path(X,Z)), N = #countY: item(to(X,Y)), N>1, current(X).

:- item(map(X,C)), item(map(Z,C)), item(map(Y,C)), dsa_edge(Y,Z), dsa_edge(X,Y), dsa_edge(X,Z),current(X), current(Y), current(Z), Y!=Z, X!=Y, X!=Z.

:- item(map(X,C)), item(map(Z,C)), item(map(Y,C)), non_dsa_edge(Y,Z), non_dsa_edge(X,Y),non_dsa_edge(X,Z), current(X), current(Y), current(Z), Y!=Z, X!=Y, X!=Z.

:- item(map(X,C)), item(map(Z,C)), item(map(Y,C)), non_dsa_edge(Y,Z), dsa_edge(X,Y), dsa_edge(X,Z),current(X), current(Y), current(Z), Y!=Z, X!=Y, X!=Z.

:- item(map(X,C)), item(map(Z,C)), item(map(Y,C)), dsa_edge(Y,Z), non_dsa_edge(X,Y), dsa_edge(X,Z),current(X), current(Y), current(Z), Y!=Z, X!=Y, X!=Z.

:- item(map(X,C)), item(map(Z,C)), item(map(Y,C)), dsa_edge(Y,Z), dsa_edge(X,Y), non_dsa_edge(X,Z),current(X), current(Y), current(Z), Y!=Z, X!=Y, X!=Z.

91

:- item(map(X,C)), item(map(Z,C)), item(map(Y,C)), non_dsa_edge(Y,Z), non_dsa_edge(X,Y),dsa_edge(X,Z), current(X), current(Y), current(Z), Y!=Z, X!=Y, X!=Z.

:- item(map(X,C)), item(map(Z,C)), item(map(Y,C)), dsa_edge(Y,Z), non_dsa_edge(X,Y), non_dsa_edge(X,Z),current(X), current(Y), current(Z), Y!=Z, X!=Y, X!=Z.

:- item(map(X,C)), item(map(Z,C)), item(map(Y,C)), non_dsa_edge(Y,Z), dsa_edge(X,Y), non_dsa_edge(X,Z),current(X), current(Y), current(Z), Y!=Z, X!=Y, X!=Z.:- N= #countY: item(to(X,Y)), N>2, current(X).

#show item/1.

#show extend/1.

92

Appendix D

CPLEX OPL implementation

D.1 Data file

L = 4 ;P = 2 ;n = 5 ;EdgesE = { <1 ,2 > ,<1 ,3 > ,<2 ,3 > ,<3 ,4 > ,<2 ,4 > ,<4 ,5 >};EdgesF = { <2 ,3 > ,<4 ,5 >};

D.2 GFBPNEC

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ DATA∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

i n t L = . . . ; / / number o f c o l o r si n t P = . . . ; / / maximum l e n g t h o f a p a t h i n a c o l o ri n t n = . . . ; / / number o f nodes i n a graph

r a n g e c o l o r s = 0 . . L−1;r a n g e l _ p a t h = 0 . . P−1;

/ / nodes and edgesr a n g e Nodes = 1 . . n ;

t u p l e edge{

i n t u ;i n t v ;

}

{ edge } EdgesE = . . . ; / / edges o f t h e g raph

93

{ edge } EdgesF = . . . ; / / edges t h a t can be a s s i g n e d t o a p a t h i n a c o l o r

{ edge } EdgesRevF = {<v , u> | <u , v> i n EdgesF } ;{ edge } EdgesDirF = EdgesF un ion EdgesRevF ;{ i n t } NbsF [ u i n Nodes ] = {v | <u , v> i n EdgesDirF } ;

/ / D e c i s i o n v a r i a b l e s/ / I f I use x [ EdgesF ] [ c o l o r s ] [ l _ p a t h ] t o d e c l a r e t h e v a r i a b l e s x_uv and x_vu f o r each edge uv \ i n F/ / t h e n x w i l l be d e f i n e d on ly f o r t h e edges i n t h e s p e c i f i c d i r e c t i o n u t o v/ / t o c o n s i d e r bo th uv and vu we use t h e s e t EdgesDirF

dva r b o o l e a n x [ EdgesDirF ] [ c o l o r s ] [ l _ p a t h ] ;dva r b o o l e a n y [ Nodes ] [ Nodes ] [ c o l o r s ] ;dva r b o o l e a n xx [ EdgesF ] [ c o l o r s ] ;dva r b o o l e a n z [ Nodes ] [ c o l o r s ] ;dva r b o o l e a n lambda [ c o l o r s ] ;

/ / s e t a t i m e l i m i t t o 3600 s e ce x e c u t e {

c p l e x . t i l i m = 3600 ;c p l e x . m i p d i s p l a y = 4 ;

}

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗MODEL∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

min imizesum ( i i n c o l o r s )

lambda [ i ] ;

s u b j e c t t o {

/ / c s t 1f o r a l l ( i i n c o l o r s )

f o r a l l ( v i n Nodes )f o r a l l ( k i n l _ p a t h )

i f ( k >=1)sum ( u i n NbsF [ v ] ) x [ <v , u > ] [ i ] [ k ]− sum ( u i n NbsF [ v ] ) x [ <u , v > ] [ i ] [ k−1] <= 0 ;

e l s esum ( <v , u> i n EdgesDirF ) x [ <v , u > ] [ i ] [ k ] <= y [ v ] [ v ] [ i ] ;

/ / c s t 2f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesF )sum ( k i n l _ p a t h ) ( x [ <u , v > ] [ i ] [ k ] + x [ <v , u > ] [ i ] [ k ] )== xx [ <u , v > ] [ i ] ;

94

/ / c s t 3f o r a l l ( i i n c o l o r s )

f o r a l l ( v i n Nodes )y [ v ] [ v ] [ i ] +sum ( k i n 1 . . P ) ( sum ( u i n NbsF [ v ] ) x [ <u , v > ] [ i ] [ k−1] ) == z [ v ] [ i ] ;

/ / c s t 4f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesF )f o r a l l ( o i n Nodes )

y [ u ] [ o ] [ i ] + xx [ <u , v > ] [ i ] − 1 <= y [ v ] [ o ] [ i ] ;

/ / c s t 5f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesF )f o r a l l ( o i n Nodes )

y [ v ] [ o ] [ i ] + xx [ <u , v > ] [ i ] − 1 <= y [ u ] [ o ] [ i ] ;

/ / c s t 6f o r a l l ( i i n c o l o r s )

f o r a l l ( v i n Nodes )sum ( o i n Nodes ) y [ v ] [ o ] [ i ] == z [ v ] [ i ] ;

/ / c s t 7f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesE )f o r a l l ( o i n Nodes )

y [ u ] [ o ] [ i ] + sum ( r i n Nodes : r != o ) y [ v ] [ r ] [ i ] <= 1 ;

/ / c s t 8f o r a l l ( v i n Nodes )

sum ( i i n c o l o r s ) z [ v ] [ i ] == 1 ;

/ / c s t 9 . . .f o r a l l ( i i n c o l o r s ) {

f o r a l l ( k i n l _ p a t h )f o r a l l ( <u , v> i n EdgesDirF )

x [ <u , v > ] [ i ] [ k ] <= lambda [ i ] ;

f o r a l l ( <u , v> i n EdgesF )xx [ <u , v > ] [ i ] <= lambda [ i ] ;

f o r a l l ( o i n Nodes )f o r a l l ( u i n Nodes )y [ u ] [ o ] [ i ] <= lambda [ i ] ;

f o r a l l ( v i n Nodes )z [ v ] [ i ] <= lambda [ i ] ;

95

}}

}

D.3 GFBPNC

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ DATA∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

i n t L = . . . ;i n t P = . . . ;i n t n = . . . ;

r a n g e c o l o r s = 0 . . L−1;r a n g e l _ p a t h = 0 . . P−1;

/ / nodes and edgesr a n g e Nodes = 1 . . n ;

t u p l e edge{

i n t u ;i n t v ;

}

{ edge } EdgesE = . . . ;{ edge } EdgesF = . . . ;

{ edge } EdgesRevF = {<v , u> | <u , v> i n EdgesF } ;{ edge } EdgesDirF = EdgesF un ion EdgesRevF ;

{ i n t } NbsF [ u i n Nodes ] = {v | <u , v> i n EdgesDirF } ;

/ / D e c i s i o n v a r i a b l e s/ / I f I use x [ EdgesF ] [ c o l o r s ] [ l _ p a t h ] t o d e c l a r e t h e v a r i a b l e s x_uv and x_vu f o r each edge uv \ i n F/ / t h e n x w i l l be d e f i n e d on ly f o r t h e edges i n t h e s p e c i f i c d i r e c t i o n u t o v/ / t o c o n s i d e r bo th uv and vu we use t h e s e t EdgesDirF

dva r b o o l e a n x [ EdgesDirF ] [ l _ p a t h ] ;dva r b o o l e a n y [ Nodes ] [ Nodes ] [ c o l o r s ] ;dva r b o o l e a n yy [ Nodes ] [ Nodes ] ;dva r b o o l e a n xx [ EdgesF ] ;dva r b o o l e a n lambda [ c o l o r s ] ;

/ / s e t a t i m e l i m i t t o 3600 s e ce x e c u t e {

c p l e x . t i l i m = 3600 ;c p l e x . m i p d i s p l a y = 4 ;

96

}

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗MODEL∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

min imizesum ( i i n c o l o r s )

lambda [ i ] ;

s u b j e c t t o {

/ / c s t 1

f o r a l l ( v i n Nodes )f o r a l l ( k i n l _ p a t h )

i f ( k >=1)sum ( u i n NbsF [ v ] ) x [ <v , u > ] [ k ]− sum ( u i n NbsF [ v ] ) x [ <u , v > ] [ k−1] <= 0 ;

e l s esum ( <v , u> i n EdgesDirF ) x [ <v , u > ] [ k ] <= yy [ v ] [ v ] ;

/ / c s t 2f o r a l l ( <u , v> i n EdgesF )

sum ( k i n l _ p a t h ) ( x [ <u , v > ] [ k ] + x [ <v , u > ] [ k ] ) == xx [ <u , v > ] ;

/ / c s t 3f o r a l l ( v i n Nodes )

yy [ v ] [ v ] +sum ( k i n 1 . . P ) ( sum ( u i n NbsF [ v ] ) x [ <u , v > ] [ k−1] ) == 1 ;

/ / c s t 4f o r a l l ( v i n Nodes )

sum ( i i n c o l o r s ) y [ v ] [ v ] [ i ] == yy [ v ] [ v ] ;

/ / c s t 5f o r a l l ( v i n Nodes )

sum ( i i n c o l o r s ) ( sum ( o i n Nodes ) y [ v ] [ o ] [ i ] ) == 1 ;

/ / c s t 6f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesF )f o r a l l ( o i n Nodes )

y [ u ] [ o ] [ i ] + xx [ <u , v >] − 1 <= y [ v ] [ o ] [ i ] ;

97

/ / c s t 7f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesF )f o r a l l ( o i n Nodes )

y [ v ] [ o ] [ i ] + xx [ <u , v >] − 1 <= y [ u ] [ o ] [ i ] ;

/ / c s t 8f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesE )f o r a l l ( o i n Nodes )

y [ u ] [ o ] [ i ] + sum ( r i n Nodes : r != o ) y [ v ] [ r ] [ i ] <= 1 ;

/ / c s t 9f o r a l l ( i i n c o l o r s ) {

f o r a l l ( o i n Nodes )f o r a l l ( u i n Nodes )y [ u ] [ o ] [ i ] <= lambda [ i ] ;

}}

D.4 GFBPC

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ DATA∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

i n t L = . . . ;i n t P = . . . ;i n t n = . . . ;

r a n g e c o l o r s = 0 . . L−1;r a n g e l _ p a t h = 0 . . P−1;

/ / nodes and edgesr a n g e Nodes = 1 . . n ;

t u p l e edge{

i n t u ;i n t v ;

}

{ edge } EdgesE = . . . ;{ edge } EdgesF = . . . ;

{ edge } EdgesRevF = {<v , u> | <u , v> i n EdgesF } ;{ edge } EdgesDirF = EdgesF un ion EdgesRevF ;

{ i n t } NbsF [ u i n Nodes ] = {v | <u , v> i n EdgesDirF } ;

98

/ / D e c i s i o n v a r i a b l e sdva r b o o l e a n x [ EdgesDirF ] [ l _ p a t h ] ;dva r b o o l e a n y [ Nodes ] [ c o l o r s ] ;dva r b o o l e a n yy [ Nodes ] [ Nodes ] ;dva r b o o l e a n xx [ EdgesF ] ;dva r b o o l e a n lambda [ c o l o r s ] ;

/ / s e t a t i m e l i m i t t o 3600 s e ce x e c u t e {

c p l e x . t i l i m = 3600 ;c p l e x . m i p d i s p l a y = 5 ;

}

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗MODEL∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

min imizesum ( i i n c o l o r s )

lambda [ i ] ;

s u b j e c t t o {

/ / c s t 1

f o r a l l ( v i n Nodes )f o r a l l ( k i n l _ p a t h )

i f ( k >=1)sum ( u i n NbsF [ v ] ) x [ <v , u > ] [ k ]− sum ( u i n NbsF [ v ] ) x [ <u , v > ] [ k−1] <= 0 ;

e l s esum ( <v , u> i n EdgesDirF ) x [ <v , u > ] [ k ] <= yy [ v ] [ v ] ;

/ / c s t 2f o r a l l ( <u , v> i n EdgesF )

sum ( k i n l _ p a t h ) ( x [ <u , v > ] [ k ] + x [ <v , u > ] [ k ] )== xx [ <u , v > ] ;

/ / c s t 3f o r a l l ( v i n Nodes )

yy [ v ] [ v ] +sum ( k i n 1 . . P ) ( sum ( u i n NbsF [ v ] ) x [ <u , v > ] [ k−1] )== 1 ;

99

/ / c s t 4f o r a l l ( <u , v> i n EdgesF )

f o r a l l ( o i n Nodes )yy [ u ] [ o ] + xx [ <u , v >] − 1 <= yy [ v ] [ o ] ;

/ / c s t 5f o r a l l ( <u , v> i n EdgesF )

f o r a l l ( o i n Nodes )yy [ v ] [ o ] + xx [ <u , v >] − 1 <= yy [ u ] [ o ] ;

/ / c s t 6f o r a l l ( v i n Nodes )

sum ( o i n Nodes ) yy [ v ] [ o ] == 1 ;

/ / c s t 7f o r a l l ( o i n Nodes )

sum ( i i n c o l o r s ) y [ o ] [ i ] == yy [ o ] [ o ] ;

/ / c s t 8f o r a l l ( o i n Nodes )

f o r a l l ( i i n c o l o r s )y [ o ] [ i ] <= lambda [ i ] ;

/ / c s t 9f o r a l l ( <u , v> i n EdgesE )

f o r a l l ( o i n Nodes )f o r a l l ( r i n Nodes )

f o r a l l ( i i n c o l o r s )i f ( o != r )

y [ o ] [ i ] + y [ r ] [ i ]<= 3 − ( yy [ u ] [ o ] + yy [ v ] [ r ] ) ;

}

D.5 ICE

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ DATA∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

i n t L = . . . ;i n t P = . . . ;i n t n = . . . ;

r a n g e c o l o r s = 0 . . L−1;

/ / nodes and edgesr a n g e Nodes = 1 . . n ;

100

t u p l e edge{

i n t u ;i n t v ;

}

{ edge } EdgesE = . . . ;{ edge } EdgesF = . . . ;

{ edge } EdgesRevF = {<v , u> | <u , v> i n EdgesF } ;/ / { edge } EdgesDirF = EdgesF un ion EdgesRevF ;

{ i n t } NbsF [ u i n Nodes ] = {v | <u , v> i n EdgesF } ;{ i n t } NbsRevF [ u i n Nodes ] = {v | <u , v> i n EdgesRevF } ;

r a n g e S = 0 . . f t o i ( round ( n ^ ( P +2) −1) ) ;

{ i n t } x [ s i n S , i i n 1 . . P+2] = { ( ( s d i v f t o i ( round ( n ^ ( i −1 ) ) ) ) mod n ) + 1 } ;{ i n t } Sub [ s i n S ] = un ion ( i i n 1 . . P+2) x [ s , i ] ;

/ / D e c i s i o n v a r i a b l e sdva r b o o l e a n xx [ EdgesF ] [ c o l o r s ] ;dva r b o o l e a n z [ Nodes ] [ c o l o r s ] ;dva r b o o l e a n lambda [ c o l o r s ] ;

/ / s e t a t i m e l i m i t t o 3600 s e ce x e c u t e {

c p l e x . t i l i m = 3600 ;c p l e x . m i p d i s p l a y = 4 ;

}

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗MODEL∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

min imizesum ( i i n c o l o r s ) lambda [ i ] ;

s u b j e c t t o {

/ / c s t 1f o r a l l ( v i n Nodes )

sum ( i i n c o l o r s ) z [ v ] [ i ] == 1 ;

/ / c s t 2

101

f o r a l l ( i i n c o l o r s )f o r a l l ( <u , v> i n EdgesF )

z [ u ] [ i ] + z [ v ] [ i ] − 1 <= xx [ <u , v > ] [ i ] ;

/ / c s t 3f o r a l l ( i i n c o l o r s )

f o r a l l ( u i n Nodes )sum ( v i n NbsF [ u ] ) xx [ <u , v > ] [ i ]+ sum ( v i n NbsRevF [ u ] ) xx [ <v , u > ] [ i ] <= 2 ;

/ / c s t 4f o r a l l ( i i n c o l o r s )

f o r a l l ( s i n S : c a r d ( Sub [ s ] ) <= P+1)sum( <u , v> i n EdgesF : u i n Sub [ s ] && v i n Sub [ s ] )xx [ <u , v > ] [ i ] <= ( c a r d ( Sub [ s ] ) −1 ) ;

/ / c s t 5f o r a l l ( i i n c o l o r s )

f o r a l l ( s i n S : c a r d ( Sub [ s ] ) == P+2)sum( <u , v> i n EdgesF : u i n Sub [ s ] &&v i n Sub [ s ] ) xx [ <u , v > ] [ i ] <= P ;

/ / c s t 6f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesE : <u , v> n o t i n EdgesF )z [ u ] [ i ] + z [ v ] [ i ] <= 1 ;

/ / c s t 7f o r a l l ( i i n c o l o r s ) {

f o r a l l ( <u , v> i n EdgesF )xx [ <u , v > ] [ i ] <= lambda [ i ] ;

f o r a l l ( v i n Nodes )z [ v ] [ i ] <= lambda [ i ] ;

}}

D.6 IFB

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ DATA∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

i n t L = . . . ;

102

i n t P = . . . ;i n t n = . . . ;

r a n g e c o l o r s = 0 . . L−1;r a n g e l _ p a t h = 0 . . P−1;

/ / nodes and edgesr a n g e Nodes = 1 . . n ;

t u p l e edge{

i n t u ;i n t v ;

}

{ edge } EdgesE = . . . ;{ edge } EdgesF = . . . ;

{ edge } EdgesRevF = {<v , u> | <u , v> i n EdgesF } ;{ edge } EdgesDirF = EdgesF un ion EdgesRevF ;

{ i n t } NbsF [ u i n Nodes ] = {v | <u , v> i n EdgesDirF } ;

/ / D e c i s i o n v a r i a b l e sdva r b o o l e a n x [ EdgesDirF ] [ c o l o r s ] [ l _ p a t h ] ;

dva r b o o l e a n y [ Nodes ] [ Nodes ] [ c o l o r s ] ;dva r b o o l e a n xx [ EdgesF ] [ c o l o r s ] ;dva r b o o l e a n z [ Nodes ] [ c o l o r s ] ;dva r b o o l e a n lambda [ c o l o r s ] ;

/ / s e t a t i m e l i m i t t o 3600 s e ce x e c u t e {

c p l e x . t i l i m = 3600 ;c p l e x . m i p d i s p l a y = 4 ;

}

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗MODEL∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

min imizesum ( i i n c o l o r s ) lambda [ i ] ;

s u b j e c t t o {

/ / c s t 1f o r a l l ( i i n c o l o r s )

103

f o r a l l ( v i n Nodes )f o r a l l ( k i n l _ p a t h )

i f ( k >=1)sum ( u i n NbsF [ v ] ) x [ <v , u > ] [ i ] [ k ]− sum ( u i n NbsF [ v ] ) x [ <u , v > ] [ i ] [ k−1] <= 0 ;

e l s esum ( <v , u> i n EdgesDirF ) x [ <v , u > ] [ i ] [ k ] <= y [ v ] [ v ] [ i ] ;

/ / c s t 2f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesF )sum ( k i n l _ p a t h ) ( x [ <u , v > ] [ i ] [ k ] + x [ <v , u > ] [ i ] [ k ] )== xx [ <u , v > ] [ i ] ;

/ / c s t 3f o r a l l ( i i n c o l o r s )

f o r a l l ( v i n Nodes )y [ v ] [ v ] [ i ] +sum ( k i n 1 . . P ) ( sum ( u i n NbsF [ v ] ) x [ <u , v > ] [ i ] [ k−1] )

== z [ v ] [ i ] ;

/ / c s t 4f o r a l l ( v i n Nodes )

sum ( i i n c o l o r s ) z [ v ] [ i ] == 1 ;

/ / c s t 5f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesF )z [ u ] [ i ] + z [ v ] [ i ] − 1 <= xx [ <u , v > ] [ i ] ;

/ / c s t 7f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesE )i f ( <u , v> n o t i n EdgesF )

z [ u ] [ i ] + z [ v ] [ i ] <= 1 ;

/ / c s t 8 . . .f o r a l l ( i i n c o l o r s ) {

f o r a l l ( <u , v> i n EdgesF )xx [ <u , v > ] [ i ] <= lambda [ i ] ;

f o r a l l ( v i n Nodes )z [ v ] [ i ] <= lambda [ i ] ;

}

104

}

D.7 IDC

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ DATA∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

i n t L = . . . ;i n t P = . . . ;i n t n = . . . ;

r a n g e c o l o r s = 0 . . L−1;

/ / nodes and edgesr a n g e Nodes = 1 . . n ;

t u p l e edge{

i n t u ;i n t v ;

}

{ edge } EdgesE = . . . ;{ edge } EdgesF = . . . ;

{ edge } EdgesRevF = {<v , u> | <u , v> i n EdgesF } ;{ edge } EdgesDirF = EdgesF un ion EdgesRevF ;

{ i n t } NbsF [ u i n Nodes ] = {v | <u , v> i n EdgesDirF } ;

/ / D e c i s i o n v a r i a b l e sdva r b o o l e a n xx [ EdgesDirF ] [ c o l o r s ] ;dva r b o o l e a n z [ Nodes ] [ c o l o r s ] ;dva r b o o l e a n lambda [ c o l o r s ] ;

/ / s e t a t i m e l i m i t t o 3600 s e ce x e c u t e {

c p l e x . t i l i m = 3600 ;c p l e x . m i p d i s p l a y = 4 ;

}

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗MODEL∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

min imize

105

sum ( i i n c o l o r s ) lambda [ i ] ;

s u b j e c t t o {

/ / c s t 1f o r a l l ( v i n Nodes )

sum ( i i n c o l o r s ) z [ v ] [ i ] == 1 ;

/ / c s t 2f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesDirF )z [ u ] [ i ] + z [ v ] [ i ] − 1 <= xx [ <u , v > ] [ i ] ;

/ / c s t 3f o r a l l ( i i n c o l o r s )

f o r a l l ( u i n Nodes )sum ( v i n NbsF [ u ] ) xx [ <u , v > ] [ i ] <= 1 ;

/ / c s t 4f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesE : <u , v> n o t i n EdgesF )z [ u ] [ i ] + z [ v ] [ i ] <= 1 ;

/ / c s t 5f o r a l l ( i i n c o l o r s ) {

f o r a l l ( <u , v> i n EdgesDirF )xx [ <u , v > ] [ i ] <= lambda [ i ] ;

f o r a l l ( v i n Nodes )z [ v ] [ i ] <= lambda [ i ] ;

}}

D.8 IAC

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ DATA∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

i n t L = . . . ;i n t P = . . . ;i n t n = . . . ;

r a n g e c o l o r s = 0 . . L−1;

/ / nodes and edgesr a n g e Nodes = 1 . . n ;

106

t u p l e edge{

i n t u ;i n t v ;

}

{ edge } EdgesE = . . . ;{ edge } EdgesF = . . . ;

{ edge } EdgesRevF = {<v , u> | <u , v> i n EdgesF } ;{ edge } EdgesDirF = EdgesF un ion EdgesRevF ;{ i n t } NbsF [ u i n Nodes ] = {v | <u , v> i n EdgesDirF } ;

/ / D e c i s i o n v a r i a b l e sdva r b o o l e a n z [ Nodes ] [ c o l o r s ] ;dva r b o o l e a n lambda [ c o l o r s ] ;

/ / s e t a t i m e l i m i t t o 3600 s e ce x e c u t e {

c p l e x . t i l i m = 3600 ;c p l e x . m i p d i s p l a y = 4 ;

}

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗MODEL∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

min imizesum ( i i n c o l o r s ) lambda [ i ] ;

s u b j e c t t o {

/ / c s t 1f o r a l l ( v i n Nodes )

sum ( i i n c o l o r s ) z [ v ] [ i ] == 1 ;

/ / c s t 2f o r a l l ( i i n c o l o r s )

f o r a l l ( u i n Nodes )sum ( v i n NbsF [ u ] ) z [ v ] [ i ]<= z [ u ] [ i ] + (1−z [ u ] [ i ] ) ∗ c a r d ( NbsF [ u ] ) ;

/ / c s t 3f o r a l l ( i i n c o l o r s )

f o r a l l ( <u , v> i n EdgesE : <u , v> n o t i n EdgesF )z [ u ] [ i ] + z [ v ] [ i ] <= 1 ;

107

/ / c s t 4f o r a l l ( i i n c o l o r s ) {

f o r a l l ( <u , v> i n EdgesF )xx [ <u , v > ] [ i ] <= lambda [ i ] ;

f o r a l l ( v i n Nodes )z [ v ] [ i ] <= lambda [ i ] ;

}}

D.9 Naïve

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ DATA∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

i n t L = . . . ;i n t n = . . . ;i n t p = . . . ;

r a n g e c o l o r s = 0 . . L−1;

/ / nodes and edgesr a n g e Nodes = 1 . . n ;

t u p l e edge{

i n t u ;i n t v ;

}

{ edge } EdgesE = . . . ;{ i n t } P a t h s [ 1 . . p ] = . . . ;r a n g e P a t h s _ i n d = 1 . . p ;

/ / D e c i s i o n v a r i a b l e sdva r b o o l e a n x [ P a t h s _ i n d ] [ c o l o r s ] ;dva r b o o l e a n lambda [ c o l o r s ] ;

/ / s e t a t i m e l i m i t t o 3600 s e ce x e c u t e {

c p l e x . t i l i m = 3600 ;c p l e x . m i p d i s p l a y = 4 ;

}

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗MODEL∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /

108

minimizesum ( i i n c o l o r s ) lambda [ i ] ;

s u b j e c t t o {

/ / c s t 1f o r a l l ( v i n Nodes )

sum ( i i n c o l o r s ) sum ( j i n P a t h s _ i n d : v i n P a t h s [ j ] ) x [ j ] [ i ]== 1 ;

/ / c s t 2f o r a l l ( i i n c o l o r s )

f o r a l l ( p1 i n P a t h s _ i n d )f o r a l l ( p2 i n P a t h s _ i n d : p1 != p2 )

f o r a l l ( u i n P a t h s [ p1 ] )f o r a l l ( v i n P a t h s [ p2 ] )

i f ( <u , v> i n EdgesE | |<v , u> i n EdgesE )

x [ p1 ] [ i ]+ x [ p2 ] [ i ]

<= 1 ;

f o r a l l ( p i n P a t h s _ i n d )f o r a l l ( i i n c o l o r s )

x [ p ] [ i ] <= lambda [ i ] ;}

109

Résumé

Maitriser les coûts de fabrication des circuits intégrés tout en augmentant leur densité est d’une im-portance primordiale pour maintenir une rentabilité dans l’industrie du semi-conducteur. Nous nousintéressons à la fabrication des « vias ». La lithographie est utilisée pour former une disposition devias sur une plaque de silicium à l’aide d’un masque. Si deux vias sont trop proche alors ils sont ditsen conflit. L’élimination des conflits se fait par une décomposition des vias en sous-ensembles sansconflit : les vias sont ensuite formés sur la plaque de silicium en séquence, c’est du Multiple Patter-ning (MP). Comme les masques sont coûteux, il faut minimiser leur nombre. Une technique basée surl’auto-assemblage des molécules (DSA) permet de grouper des vias en conflit selon certaines condi-tions. Le but est de trouver la meilleure façon de grouper les vias afin de minimiser le nombre demasques. Nous étudions des méthodes exactes pour concevoir des solutions optimales à ce problème.

Abstract

Controlling the manufacturing costs of integrated circuits while increasing their density is of a paramountimportance to maintain a certain degree of profitability in the semi-conductor industry. Our focus is onthe fabrication of components called « vias ». The lithography is used to form an arrangement of viason a silicon wafer using an optical mask. If two vias are too close to be formed using lithography, theyare said in conflict. The elimination of conflicts is done by the decomposition of the vias in subsetswithout conflict: the vias are then formed on the silicon wafer in sequence, this is Multiple Patterning(MP). Since masks are expensive, it is important to minimize their number. A technique based on thedirection and self-assembly of molecules (DSA) can group vias in conflict according to certain condi-tions. The goal is to find the best way of grouping the vias to minimize the number of masks. We arestudying exact methods to design optimal solutions to this problem.